Solution : a 0 an 0
6 3 −4 −1 Ex. 6 : If A = 0 b , prove that An = 0 bn
−04 = [x
3
Given [2 0 3] −1 2 + 2 1 y] for all n∈N.
5
4 −3 a 0
Solution : Given A= 0
b
18 9 −8 −2 an 0
∴ [2 0 3] 1−53 −08 = [x
6 + 2 y] We prove An = 0 bn for all n∈N using
12 −6
mathematical induction
10 7 Let P(n) be = an 0 for n∈N.
∴ [2 0 3] −1 6 = [x y]
0 bn
9 4
To prove that P(n) is true for n=1
∴ [20 + 27 14 + 12] = [x y] a 0
P(1) is A1 = A= 0 b ∴ P(1) is true.
∴ [47 26] = [x y] ∴ x = 47 , y = 26 by
definition of equality of matrices. Assume that P(K) is true for some K∈N
sinθ That is P(K) is Ak = aK 0
Ex. 5 : Find if cosθ [sinθ cosθ θ] = [17] 0 b K
θ
To prove that P(K) → P(K+1) is true consider
L.H.S. of P(K+1)
sinθ That is Ak+1
cosθ
Solution : Let [sinθ cosθ θ] = [17], = Ak.A
θ aK 0 a 0 aK +1 + 0 0 + 0
∴ [sin2θ + cos2θ + θ2 ] = [17] = 0 b K 0 b = 0+0 bK +1 + 0
∴ By definition of equality of matries
∴ 1 + θ2 = 17 ∴ θ2 = 17–1 ∴ θ2 = 16, aK +1 0 = R.H.S of
= P(K+1)
0 bK +1
∴ θ = ± 4
Hence P(K+1) is true.
Remark ∴ P(K) ⇒ P(K+1) for all K∈N
Using the distributive laws discussed earlier
Hence by principle of mathematical induction,
we can derive the following results, the statement P(n) is true for all n∈N.
If A and B are square matrices of the same That is P(n) is true → P(2) is true → P(3) is true
order, then and so on → P(n)is true, n∈N.
i) (A + B)2 = A2 + AB + BA + B2 ∴ = An an 0 for all n∈N.
ii) (A – B)2 = A2 – AB – BA + B2 0 bn
iii) (A + B) (A– B)= A2 + AB – BA – B2
92
Ex. 7 : A school purchased 8 dozen Mathematics Ex. 8 : Some schools send their students for extra
books, 7 dozen Physics books and 10 dozen training in Kabaddi, Cricket and Tennis to a sports
Chemistry books, the prices are Rs.50,Rs.40 standidium. There center charge fee is changed
and Rs.60 per book respectively. Find the total pen student for Coaching as well as 4 equipment
amount that the book seller will receive from and maintenances of the court. The information of
school authority using matrix multiplication. students from each school is given below-
Solution : Let A be the column matrix of books
of different subjects and let B be the row matrix Modern School Kabaddi Cricket Tennis
of prices of one book of each subject. 20 35 15
Progressive School 18
36 12
8×12 96 Sharada Sadan
perstud22 e45nt 8
A= 7 ×12 = 84 B = [50 40 60] Vidya Niketan 12 game6are
The charges ea2c0h
10×12 120
for
∴ The total amount received by the given below-
bookseller is obtained by matrix BA. E and M is for
∴ BA = [50 40 60] 96 Kabaddi Coa ch E & M equipment and
maintain
84 Cricket 40 10
120 Tennis 50 50
60 40
= [50 × 96 + 40 × 84 + 60 × 120]
= [4800 + 3360 + 7200] = [15360]
Thus the amount received by the bookseller
from the school is Rs.15360.
To find the expense of each school on Coaching and E and M can be fourd by multiplication of the above
matrics.
Kabaddi Cricket Tennis Coach E & M
Modern 20 35 18 Kab 40 10
Progreesive 18 36 12 X cri 50
Sharda School 24 50
Vidya Niketan 25
12 8 Ten 60 40
20 6
Modern 20 × 40 + 36 × 50 + 18 × 60 20 × 10 + 35 × 50 + 18 × 40
Progressive 18 × 40 + 36 × 50 + 12 × 60 18 × 10 + 36 × 50 + 12 × 40
Sharada 24 × 40 + 12 × 50 + 8 × 60 24 × 10 + 12 × 50 + 8 × 40
Vidya 25 × 40 + 20 × 50 + 6 × 60 25 × 10 + 20 × 50 + 6 × 40
800 + 1750 + 108 200 + 1750 + 720 2650 2670
720 + 1800 + 720 180 + 1800 + 480 = 3240 2960
= 960 + 600 + 480 240 + 600 + 320 2040 1160
1000 + 1000 + 360 250 + 1000 + 240 2360 1490
93
EXERCISE 4.6 2 4 3 2 −2
ii) A = −1 2 , B= and
3 3 3
3 4 −1 1
1) Evaluate i) 2[2 −4 3] ii)[2 −1 3] 3 3 1
C = 1 3 .
1 1
1 −3 ,B= 4 1 7) Verify that A(B+C)=AB+BC in each of the
2) If A = 4 3 −2 show that following matrices
2 4 −2 −1 1
AB ≠ BA. i) A = 2 3 , B= 3 −2 and
−1 1 1 4 1
3 0 , C = 2 −1
3) If A= 2
1 −3 1
2 1 4 1 −1 3 1 0
B = 3 0 2 . State whether AB=BA? ii) A = 2 3 2 , −2 3 and
B= 4 3
1 2 1
1 2
Justify your answer. C = −2
0
4) Show that AB=BA where, 4 −3
−2 3 −1 1 3 −1 1 −2 3 −1
i) A = −1 2 −1 , B = 2 2 −1
8) If A = 5 6 , B= 3 7 ,
−6 9 −4 3 0 −1
Find AB-2I,where I is unit matrix of order 2.
ii) A= cosθ sinθ , B= cos sin
cos
sinθ cosθ sin 4 3 2 1 2
9) If A = −1 2 0
−1
,B= 1 0 show
4 8 −2
5) If A = −2 −4 , prove that A2 = 0.
that matrix AB is non singular.
6) Verify A(BC) = (AB)C in each of the 1 2 0
following cases. 10) If A = 5 4
2 , find the product
1 0 1 2 −2 0 7 −3
i) A = 2 3 0 , B = −1 1 and (A+I)(A−I).
0 4 5 0 3
3 2 −1 ∝ 0 1 0
C = 2 0 −2 11) A= 1 , B = 2 1 find α, if A2 = B.
1
94
1 2 2 21) Find x and y, if
12) If A= 2 1 2 , Show that A2–4A is a
2 1 2 −1 3 − 3 −3 4 2 = x
2 4 1 0 2 2 1 1 −1 y
1
scalar matrix. 22) Find x, y, z if
13) If A = 1 0 , find k so that 2 0 1 1 1 x − 3
−1 7 3 0 2 − 4 −1 2 2
2 2 3 1 = y −1 .
2z
A2 –8A –kI = O, where I is a unit matrix and
O is a null matrix of order 2.
cosα sinα show
23) If A = −sinα , that
8 4 5 −4 cosα
14) If A = 10 5 , B = 10 −8 show that
cos2α sin2α
(A + B)2 = A2 + AB + B2. A2 = −sin2α cos2α .
3 1 1 2 0 4
15) If A = −1 2 , prove that A2 – 5A + 7I=0, 24) If A = 3 5 , B = 2 −1 , show that
where I is unit matrix of order 2.
AB ≠ BA, but |AB| = |A|.|B|
3 4 2 1 25) Jay and Ram are two friends in a class. Jay
16) If A = −4 3 and B = −1 2 , show that wanted to buy 4 pens and 8 notebooks, Ram
wanted to buy 5 pens and 12 notebooks.
(A+B)(A-B)= A2– B2. Both of them went to a shop. The price
of a pen and a notebook which they have
1 2 2 a selected was Rs.6 and Rs.10. Using Matrix
17) If A = −1 −2 , B = −1 b and if multiplication, find the amount required
from each one of them.
(A + B)2 = A2+B2. find values of a and b.
4.7 Properties of transpose of a matrix :
18) Find matrix X such that AX=B, where Note :
1 −2 and B −3 (1) For any matrix A, (AT)T = A.
A= −2 = −1 .
1 (2) If A is a matrix and k is constant,
then (kA)T = kAT
3 −2 (3) If A and B are two matrices of same order,
19) Find k, if A= 4 −2 and if A2 = kA-2I. then (A + B)T = AT + BT
1 2 3 1 (4) If A and B are conformable for the product
20) Find x, if [1 x 1] 4 5 6 −2 = 0.
AB, then (AB)T = BT AT
3 2 5 3
1 2 1 2 3
Example, Let A = 3 1 3 , 1 2 ,
B = 1 2
95
∴ AB is defined and (7) If A is a square matrix, then (a) A + AT is
2 + 2 +1 3+ 4 + 2 5 9 symmetric. (b) A – AT is skew symmetric.
AB = 6 +1+ 3 9 + 2 + 6 = 10 17 , 3 5 7
For example, (a) Let A = 2 4 −6 ,
5 10
∴ (AB)T = 9 17 ……… (1) 3 8 −5
1 3 3 2 3
2 1 1 ∴ AT = 5 4
Now AT = 2 1 , BT = 3 2 2 , 8
1 3
7 −6 −5
2 1 1 1 3 3 5 7 3 2 3
∴ BT AT = 3 2 2 2 1
1 3 Now A + AT = 2 4 −6 + 5 4 8
3 8 −5 7 −6 −5
2+ 2+1 6+1+3 6 7 10
∴ BT AT = 3 + 4 + 2 9 + 2 + 6
= 7 8 2
5 10 10 2 −10
= 9 17 …….. (2)
∴ From (1) and (2) we have, (AB)T = BT AT ∴ A + AT is a symmetric matrix, by definition.
In general (A1 A2 A3, ....... An)T = AnT .... A3T A2T
A1T 3 5 7 3 2 3
(b) Let A – AT = 2 4 −6 – 5 4
8
(5) If A is a symmetric matrix, then AT = A. 3 8 −5 7 −6 −5
2 −3 4
0 3 4
For example, let A = −3 5 −2 = −3 0 −14
4 −2 1
−4 14 0
2 −3 4
AT = −3 5 −2 = A A − AT is a skew symmetric matrix, by definition.
4 −2 1 Note:
(6) If A is a skew symmetric matrix, then A square matrix A can be expressed as the
AT = – A. sum of a symmetric and a skew symmetric matrix
0 5 4 as A = 1 (A + AT) + 1 (A –AT )
For example, let A = −5 0 −2 2 2
−4 2 0 4 −5 3
e.g. Let A = −6 2
1 ,
0 −5 −4 0 5 4 7 8 −9
∴ AT = 5 0 2 = − −5 0 −2 4 −6 7
4 −2 0 −4 2 0 ∴ AT = −5 2
8
= –A, ∴ AT = –A 3 1 −9
96
4 −5 3 4 −6 7 EXERCISE 4.7
A + AT = −6 2 −5
1 + 2 8
7 8 −9 3 1 −9 (1) Find AT, if (i) A = 1 3
−4 5
8 −11 10
= −11 4
9 2 −6 1
(ii) A = −4 0 5
10 9 −18
1 1 8 −11 10 (2) If [ aij]3x3 where aij = 2(i–j). Find A and AT.
2 2 −11 State whether A and AT are symmetric or
Let P = (A + AT) = 10 4 9
9 skew symmetric matrices ?
−18
4 − 11 5 5 −3
2 −3 , Prove that (2A)T = 2AT.
2 1
= − 11 9 (3) If A = 4
2
2 −2
9 −9
5 2 1 2 −5
The matrix P is a symmetric matrix. (4) If A = 2 −3 4 , Prove that (3A)T
−5 4 9
4 −5 3 4 −6 7 = 3AT.
Also A – AT = −6 2 −5
1 – 2 8
7 8 −9 3 1 −9 0 1+ 2i i − 2
(5) If A = −1− 2i 0
0 1 −4 7 −7
= −1 0 −7 2 − i
0
4 7 0
where i = -1,,Prove that AT = –A.
1 1 0 1 −4 2 −3 2 1
2 2 −1 −7
Let Q = (A – AT) = 4 0 0 (6) If A = 5 −4 , B = 4 −1 and
7
−6 1 −3 3
0 1 −2 1 2
2 C = −1 4 then show that
= − 1 0 − 7 −2 3
2 2
7 (i) (A + B) = AT + BT (ii) (A – C)T = AT – CT
2 2 0
5 4 and B = −1 3
The matrix Q is a skew symmetric matrix. (7) If A = −2 3 −1 , then find
4
Since P+Q = symmetric matrix + skew symmetric
matrix. CT, such that 3A – 2B + C = I, where I is the
unit matrix of order 2.
Thus A = P + Q.
97
7 3 0 0 −2 3 Let's Remember
(8) If A 0 4 −2 , B = 2 1 −4 then
find (i) AT + 4BT (ii) 5 AT – 5BT. • The value of a determinant of order 3 × 3
1 0 1 2 1 −4 a1 b1 c1
(9) If A = 3 1 2 , B = 3 5 −2 and a2 b2 c2 = a1(b2c3–b3c2)–b1(a2c3–a3c2)+
a3 b3 c3 c1(a2b3– a3b2)
0 2 3 • The minors and cofactors of elements of a
C = −1 −1 0 , verify that determinant
(A + 2B + 2C)T = AT + 2BT + 3CT.
∆ = a11 a12 a13
a21 a22 a23
−1 2 1 2 1 a31 a32 a33
(10) If A = −3 2 −3 −3 2 , prove
and B = −1 3
that (A + BT )T = AT + B. Minor Mij of the element aij is determinant
obtained by deleting the ith row and jjth column of
(11) Prove that A + AT is a symmetric and A – AT determinant D. The cofactor Cij of element aij is
is a skew symmetric matrix, where Cij = (–1)i+j Mij
1 2 4 5 2 −4 • Properties of determinant
1 (ii) A = 3 −7
(i) A = 3 2 2 Property (i) - The value of determinant remains
unchanged if its rows are turned into columns
−2 −3 2 4 −5 −3 and columns are turned into rows.
(12) Express the following matrices as the sum of Property (ii) - If any two rows (or columns) of
a symmetric and a skew symmetric matrix. the determinant are interchanged then the value
of determinant changes its sign.
3 3 −1
4 −2 −2 −2
(i) 3 −5 (ii) −4 −5 1 Property (iii) - If any two rows (or columns)
of a determinant are identical then the value of
2 determinant is zero
2 −1 0 3 −4
(13) If A = 3 −2 and B = 2 −1 1 ,
4 1 Property (iv) - If any element of a row (or column)
of determinant is multiplied by a constant k then
verify that (i) (AB)T = BT AT the value of the new determinant is k times the
value of old determinant
(ii) (BA)T = AT BT
cosα sinα , show that AT A = I, Property (v) - If each element of a row (or
(14) If A = −sinα cosα column) is expressed as the sum of two numbers
then the determinant can be expressed as sum of
where I is the unit matrix of order 2. two determinants
98
Property (vi) - If a constant multiple of all • Addition of matrices:
elements of a row (or column) is added to the Matrices A = [aij] and B = [bij] are said to
corresponding elements of any other row (or
column) then the value of new determinant conformable for addition if orders of A and B
so obtained is the same as that of the original are same.
determinant.
A+B = [aij + bij] . The order of A+B is the same
as that of A and B.
Property (vii) - (Triangle property) - If each • Multiplication of two matrices:
element of a determinant above or below the main A and B are said to be conformable for the
diagonal is zero then the value of the determinant
is equal to the product of its diagonal elements. multiplication if number of columns of A is
equal to the number of rows of B.
• A system of linear equations, using Cramer’s That is If A = [aik]m×p and B = [bkj]p×n, then AB
Rule has solution -
is defined and AB = [cij]m×n where
D=x , y Dy Dz
=x D D ,z = D ; provided D≠0 n i =1,2,….,m
j =1,2,…..,n.
• Consistency of three equations. ∑ cij = aik .bkj
a1x +b1y+c1 = 0 k =1
a2x +b2y+c2 = 0
• If A = [aij]m×n is any matrix, then the transpose
a3x +b3y+c3 = 0 are consistent of A is denoted by AT = B = [bij]m×n and bji = aij
a1 b1 c1 • If A is a square matrix, then
Then a2 b2 c2 = 0
i) A + AT is a symmetric matrix.
a3 b3 c3
ii) A − AT is a skew-symmetric matrix.
• Area of triangle whose vertices are (x1, y1), • Every square matrix A can be expressed as
(x2,y2) (x3,y3) is the sum of a symmetric and skew-symmetric
matrix as
1 x1 y1 1 11
A (∆) = 2 x2 y2 1 A= 2 [A + AT] + 2 [A – AT].
x3 y3 1 MISCELLANEOUS EXERCISE - 4 (B)
• Test for collinear of points (x1,y1), (x2,y2), (I) Select the correct option from the given
alternatives.
x1 y1 1 1 3 1 0
y2 1 = 0
(x3,y3) if x2 y3 1 1) Given A = 2 2 , I = 0 1 if A–λI is a
x3 singular matrix then ……….
A) λ = 0 B) λ2 – 3λ – 4 = 0
• Multiplication of a matrix by a scalar: C) λ2 + 3 – 4 = 0 D) λ2 – 3λ – 6 = 0
If A [aij] is a matrix and k is a scalar, then kA =
[kaij].
99
4 6 −1 7 4 1 2
2) Consider the matrices A = 3 0
2 , 7) If A + B = 8 9 and A – B =0 3
1 −2 5 then the value of A is …….
2 4 3
1 , C =1 out of the given
B = 0 3 1 4 3
A) 4 3 B) 4 6
−1 2 2
6 2 7 6
matrix product ………... C) 8 6 D) 8 12
i) (AB)TC ii) CTC(AB)T
iii) CTAB iv) ATABBTC x 3x − y 3 2
A) Exactly one is defined 8) If zx + z 3y − w = 4 7 then ……….
B) Exactly two are defined
C) Exactly three are defined
D) all four are defined a) x = 3, y = 7, z = 1, w = 14
a) x = 3, y = −5, z = −1, w = −4
3) If A and B are square matrices of equal a) x = 3, y = 6, z = 2, w = 7
order, then which one is correct among the a) x = −3, y = –7, z = –1, w = –14
following?
9) For suitable matricesA, B, the false statement
A) A + B = B + A B) A + B = A – B is …….
C) A – B = B – A D) AB = BA
A) (AB)T = ATBT
1 2 2 B) (AT)T = A
4) If A = 2 1 −2 is a matrix satisfying the C) (A − B)T = AT − BT
D) (A + B)T = AT + BT
a 2 b
equation AAT = 9I, where I is the identity −2 1
matrix of order 3, then the ordered pair (a, b) 10) If A = 0 3 and f(x) = 2x2 – 3x, then
is equal to ……. f(A) = ………
A) (2, –1) B) (–2, 1) 14 1 −14 1
C) (2, 1) D) (–2, –1)
α 2 A) 0 −9 B) 0 9
5) If A = 2 α and |A3| = 125,
then α = …….
14 −1 −14 −1
A) ±3 B) ±2 C) ±5 D) 0 C) 0 9 D) 0 −9
5 7 1 2 4 5 (II) Answer the following question.
−3 5 = 4 −4 then ……
6) If x 1 – 1) If A = diag [2 –3 –5], B = diag [4 –6 –3]
and C= diag [–3 4 1] then find i) B + C – A
2 6 2 y 0 4 ii) 2A + B – 5C.
A) x = 1, y = –2 B) x = –1, y = 2
C) x = 1, y = 2 D) x = –1, y = –2
100
cosα −sinα 0 1 −1 0 2 2 −4
0 , Find
2) If f (α) =A= sinα cosα 8) If A= 2 3 4 , B= −4 2 −4 ,
0 0 1 0 1 2 2 −1 5
i) f(–α) ii) f(–α) + f(α). show that BA=6I.
3) Find matrices A and B, where 2 1 1 2
9) If A = 0 3 , B= 3 −2 , verify that
1 −1 1 −1 |AB|= |A||B|.
i) 2A – B= 0 1 and A +3 B= 0 1
−1 2 1 cosα sinα
ii) 3A – B= 0 5 and 10) If Aα = −sinα cosα ,
1 show that Aα. Aβ = Aα+β.
0 0 1 1 ω ω2 1
A + 5B= −1 0 0
11) If A = ω2 1 , B = 1 ω , where is
2 −3
−2 , −3 4 1 a complex cube root of unity, then show that
4) If A= 3 4 B = −1 −3 AB+BA+A−2B is a null matrix.
2
−1
Verify i) (A + BT)T = AT + 2B. 2 −2 −4
12) If A = −1 3
ii) (3A – 5BT)T = 3AT – 5B. 4 show that A2 =A.
1 −2 −3
5) If A = cosα −sinα and A + AT = I, where 4 1 4
13) If A = 3 0 4 , show that A2 = I.
sinα cosα 1 3
3
I is unit matrix 2 × 2, then find the value of α.
1 2 1 3 2 3 −5 that A2 – 5A – 14I =
2 4 −1 −3 , show 14) If A = −4 , show 0.
6) If A= 3 0 and B= 2
−1
that AB is singular. 2 −1 that A2–
15) If A = −1 , show 4A + 3I = 0.
2
1 2 3 1 −1 1
7) If A = 2 4 6 , B= −3 2 −1 , show
16) −3 2 1 x
If A = −4 , B = 0 , and
1 2 3 −2 1 0 2 y
that AB and BA are both singular matrices. (A + B)(A – B) = A2 – B2, find x and y.
101
0 1 0 −1 2 −4 1 −1 2
17) If A= 1 0 and B = 1 show that 22) If A = 3 −2 −2 1 0 ,
0 0 1 , B =
(A + B)(A – B) ≠ A2 – B2. show that (AB)T = BT AT.
2 −1 3 −4
18) If A = 3 −2 , find A3. 23) If A = 1 −1 , prove that
19) Find x, y if, 1+ 2n −4n
An = 1− 2n , for all n∈N.
n
4 5 4 3
2 3 1 −41
i) [0 –1 4] 2 6 + 3 cosθ sinθ
24) If A = −sinθ cosθ , prove that
−1 0
= [x y]. cosnθ sinnθ
ii) An = −sinnθ conθ , for all n∈N.
−112 2 1 2 −3 7 5 x 25) Two farmers Shantaram and Kantaram
0 3 1 −1 3 y cultivate three crops rice,wheat and
+ 3 0 = groundnut. The sale (In Rupees) of these
crops by both the farmers for the month of
−1 April and may 2008 is given below,
20) Find x, y, z if April sale (In Rs.)
0 1 2 1 2 x −1
5 1 0 − 3 3 −2 1 y +1
i) 1 1 1 3 = 2z Rice Wheat Groundnut
Shantaram 15000 13000 12000
Kantaram 18000 15000 8000
1 3 2 3 0 2 1 x
ii) 32 0 1 + 2 1 4 05 2 = y May sale (In Rs.)
1 2 2 1
3 z Rice Wheat Groundnut
Shantaram 18000 15000 12000
2 1 −3 1 0 −2 Kantaram 21000 16500 16000
21) If A = 0 2 6 , B = 3 −1 4 , Find
find ABT and ATB. i) The total sale in rupees for two months of
each farmer for each crop.
ii) the increase in sale from April to May for
every crop of each farmer.
102
5 Straight Line
Let's Study Let's Learn
• Locus of a points in a co-ordinate plane 5.1 Locus : A set of points in a plane
• Equations of line in different forms which satisfy certain geometrical condition
• Angle between two lines, perpendicular (or conditions) is called a locus.
L = {P | P is a point in the plane and P
and parallel lines
• Distance of a point from a line satisfies given geometrical condition}
• Family of lines Here P is the representative of all points in L.
L is called the locus of point P. Locus is a
Let's Recall set of points.
The locus can also be described as the route of
We are familiar with the properties of a point which moves while satisfying required
straight lines, the bisector of an angle, circle conditions. eg. planets in solar system.
and triangles etc. Illustration :
• The perpendicular bisector of segment AB
We will now introduce co-ordinate
geometry in the study of a plane. Every point is the set
has got pair of co-ordinates and every pair of M = { P | P is a point such that PA=PB}.
co-ordinate gives us a point in the plane. • The bisector of angle AOB is the set :
D = { P | P is a point such that
We will use this and study the curves with
the help of co-ordinates of the points. P is equidistant from OA and OB }
What is the perpendicular bisector of a = { P | ∠ POA = ∠ POB }
segment ? ............... A line. Verify that the sets defined above are the same.
The plural of locus is loci.
What is the bisector of an angle ? .......A ray.
Fig. 5.1
These geometrical figures are sets of points • The circle with center O and radius 4 is
in plane which satisfy certain conditions.
• The perpendicular bisector of a segment is the set L = { P | OP = 4}
the set of points in the plane which are
equidistant from the end points of the
segment. This set is a line.
• The bisector of an angle is the set of points
in the plane which are equidistant from the
arms of the angle. This set is a ray.
Activity : Draw segment AB of length 6 cm.
Plot a few points which are equidistant from
A and B. Verify that they are collinear.
103
5.1.1 Equation of Locus : If the set of points, ∴ PA = PB
whose co-ordinates satisfy a certain equation in ∴ PA2 = PB2
x and y, is the same as the set of points on ∴ (x + 3)2 + (y − 0)2 = (x − 3)2 +
a locus, then the equation is said to be the
equation of the locus. (y −0)2
∴ x2 + 6x + 9 + y2 = x2 − 6x + 9 + y2
SOLVED EXAMPLES ∴ 12x = 0
∴ x = 0. The locus is the Y-axis.
Ex.1 We know that the y co-ordinate of every
point on the X-axis is zero and this is true 5.1.2 Shift of Origin : Let O′(h, k) be a point
for points on the X-axis only. Therefore in the XY plane and the origin be shifted to
the equation of the X-axis is y = 0. O′. Let O′X′, O′Y′ be the new co-ordinate
axes through O′ and parallel to the axes OX
Ex.2 Let L = {P | OP = 4}. Find the equation and OY respectively.
of L.
Fig. 5.2
Solution : L is the locus of points in the Fig. 5.3
plane which are at 4 unit distance from
the origin. Let (x, y) be the co-ordinates of P referred to
Let P(x, y) be any point on the locus L. the co-ordinates axes OX, OY and (x', y') be
As OP = 4, OP2 = 16 the co-ordinates of P referred to the co-ordinate
∴ (x − 0)2 + (y − 0)2 = 16 axes O′X′, O′Y′. To find relations between
∴ x2 + y2 = 16 (x, y) and (x', y').
This is the equation of locus L.
The locus is seen to be a circle Draw PL ⊥ OX and suppose it intersects O′X′
in L′.
Ex.3 Find the eqation of the locus of points
which are equidistant from A(−3, 0) and Draw PM ⊥ OY and suppose it intersects O′Y′
B(3, 0). Identify the locus. in M′.
Solution : Let P(x, y) be any point on the Let O′Y′ meet line OX in N and O′X′ meet
required locus. OY in T.
P is equidistant from A and B. ∴ ON = h, OT = k, OL = x, OM = y,
O′L′ = x', O′M′ = y'
Now x = OL = ON + NL = ON + O′ L′
104
= h + x' Ex. 3) Obtain the new equation of the locus
x2 − xy − 2y2 − x + 4y + 2 = 0 when the
and y = OM = OT + TM = OT + O′ M′ = origin is shified to (2, 3), the directions of the
k + y' axes remaining the same.
∴ x = x' + h, y = y' + k
These equations are known as the formulae for Solution : Here (h, k) = (2, 3) and if new
shift of origin. co-ordinates are (X,Y).
Note that the new co-ordinates can also be ∴ x = X + h, y = Y + k gives
given by (X, Y) or (u,v) in place of (x',y').
∴ x = X + 2, y = Y + 3
SOLVED EXAMPLES
The given equation
Ex. 1) If the origin is shifted to the point x2 − xy − 2y2 − x + 4y + 2 = 0 becomes
O'(3, 2) the directions of the axes remaining (X+2)2 − (X + 2)(Y + 3) − 2(Y + 3)2 −
the same, find the new co-ordinates of the (X + 2) + 4(Y + 3) + 2 = 0
points ∴ X2 − XY − 2Y2 − 10Y − 8 = 0
(a)A(4, 6) (b) B(2,−5). This is the new equation of the given locus.
Solution : We have (h, k) = (3,2)
EXERCISE 5.1
x = x' + h, y = y' + k 1. If A(1,3) and B(2,1) are points, find the
∴ x = x' + 3. and y = y' + 2 .......... (1) equation of the locus of point P such that
(a) (x, y) = (4, 6) PA = PB.
∴ From (1), we get 4 = x1 + 3, 6 = 2. A(−5, 2) and B(4, 1). Find the equation of
y1 + 2 the locus of point P, which is equidistant
from A and B.
∴ x' = 1 and y' = 4.
3. If A(2, 0) and B(0, 3) are two points, find
New co-ordinates of A are (1, 4) the equation of the locus of point P such
that AP = 2BP.
(ii) (x, y) = ( 2,-5)
4. If A(4, 1) and B(5, 4), find the equation
from (1), we get 2 = x' + 3, −5 = y' + 2 of the locus of point P if PA2 = 3PB2.
∴ x' = −1 and y' = −7. New co-ordinates of
B are (−1,−7) 5. A(2, 4) and B(5, 8), find the equation of
the locus of point P such that
Ex. 2) The origin is shifted to the point
(−2, 1), the axes being parallel to the original PA2 − PB2 = 13.
axes. If the new co-ordinates of point A are
(7, −4), find the old co-ordinates of point A. 6. A(1, 6) and B(3, 5), find the equation of
the locus of point P such that segment
Solution : We have (h, k) = (−2, 1) and if AB subtends right angle at P. (∠APB
new co-ordinates are (X,Y). = 90°)
x = X + h, y = Y + k 7. If the origin is shifted to the point
O′(2, 3), the axes remaining parallel to the
∴ x = X − 2, y = Y +1 original axes, find the new co-ordinates of
the points
(X, Y) = (7, −4)
(a) A(1, 3) (b) B(2, 5)
we get x = 7 − 2 = 5, y = −4 + 1 = − 3.
∴ Old co-ordinates A are (5, −3)
105
8. If the origin is shifted to the point Let's Think :
O′(1, 3) the axes remaining parallel to the
original axes, find the old co-ordinates of • Can intercept of a line be zero ?
the points
• Can intercept of a line be negative ?
(a) C(5, 4) (b) D(3, 3)
5.1.2 Inclination of a line : The smallest angle
9. If the co-ordinates A(5, 14) change made by a line with the positive direction of
to B(8, 3) by shift of origin, find the the X-axis measured in anticlockwise sense is
co-ordinates of the point where the origin called the inclination of the line. We denote
is shifted. inclination by θ. Clearly 0° ≤ θ < 180°.
10. Obtain the new equations of the following Remark : Two lines are parallel if and only
loci if the origin is shifted to the point if they have the same inclination.
O'(2, 2), the direction of axes remaining
the same : The inclination of the X-axis and a line
parallel to the X-axis is Zero. The inclination
(a) 3x − y + 2 = 0 of the Y-axis and a line parallel to the Y-axis
is 900.
(b) x2 + y2 − 3x = 7
5.2.2 Slope of a line : If the inclination of
(c) xy − 2x − 2y + 4 = 0 a line is θ then tanθ (if it exist) is called
the slope of the line. We denote it by m.
(d) y2 − 4x − 4y + 12 = 0 ∴ m = tanθ.
5.2 Straight Line : The simplest locus in a Activity : If A(x1, y1), B(x2, y2) are any two
plane is a line. The characteristic property of points on a non-vertical line whose inclination
this locus is that if we find the slope of a
segment joining any two points on this locus, is θ then verify that
then the slope is constant.
If a line meets the X-axis in the point tan θ = y2 − y1 , where x1 ≠ x2.
A (a, 0), then 'a' is called the X-intercept of x2 − x1
the line. If it meets the Y-axis in the point
B(0, b) then 'b' is called the Y-intercept of The slope of the Y-axis is not defined.
the line. Similarly the slope of a line parallel to the
Y-axis is not defined. The slope of the X-axis
is 0. The slope of a line parallel to the X-axis
is also 0.
Remark : Two lines are parallel if and only
if they have the same slope.
SOLVED EXAMPLES
Fig. 5.4 Ex. 1) Find the slope of the line whose
inclination is 60°.
Remarks :
(1) A line parallel to X-axis has no X-intercept. Solution : The tangent ratio of the inclination
(2) A line parallel to Y-axis has no Y-intercept. of a line is called the slope of the line.
Inclination θ = 60°.
∴ slope = tanθ = tan60° = 3 .
106
Ex. 2) Find the slope of the line which passes α − β = 900 or α − β = −900
through the points A(2, 4) and B(5, 7). α − β = ± 900
∴ cos(α − β ) = 0
Solution : The slope of the line passing ∴ cosα cosβ + sinα sinβ = 0
∴ sinα sinβ = −cosα cosβ
through the points (x1, y1) and (x2, y2) is given ∴ tanα tanβ = −1
by ∴ m1m2 = −1
y2 − y1
x2 − x1
y2 − y1
Slope of the line AB = x2 − x1 = 7−4 =1
5−2
Note that x1 ≠ x2.
Ex. 3) Find the slope of the line which passes
through the origin and the point A(- 4, 4).
Solution : The slope of the line passing
through the points (x1, y1) and (x2, y2) is given
by y2 − y1
x2 − x1
m= . Fig. 5.5 Fig. 5.6
Here A(−4, 4) and O(0, 0).
Slope of the line OA = y2 − y1 = 0 − 4 = −1. SOLVED EXAMPLES
x2 − x1 0 + 4
Ex. 1) Show that line AB is perpendicular to
Note that x1 ≠ x2. line BC, where A(1, 2), B(2, 4) and C(0, 5).
5.2.3 Perpendicular Lines : We know that Solution : Let slopes of lines AB and BC be
the co-ordinate axes are perpendicular to each m1 and m2 respectively.
other. Similarly a horizontal line and a vertical
line are perpendicular to each other. Slope of ∴ m1 = 4−2 =2 and
one of them is zero whereas the slope of the 2 −1
other one is not defined.
y2 − y1 = 5 − 4 = − 1
Let us obtain a relation between slopes of non- m2 = x2 − x1 0 − 2 2
vertical lines.
Now m1 × m2 = 2 × − 1 = −1
2
Theorem : Non-vertical lines having slopes m1
and m2 are perpendicular to each other if and ∴ Line AB is perpendicular to line BC.
only if m1×m2= −1.
Ex. 2) A(1,2), B(2,3) and C(−2,5) are vertices
Proof : Let α and β be inclinations of lines of ∆ ABC. Find the slope of the altitude drawn
having slopes m1 and m2. As lines are non from A.
vertical α ≠ π and β ≠ π
Solution : The slope of line BC is
22
m1 = y2 − y1 = 5 − 3 = − 2 = − 1
∴ tanα = m1 and tanβ = m2 x2 − x1 −2 − 2 4 2
From Fig. 5.5 and 5.6 we have, Altitude drawn from A is perpendicular to
BC.
107
If m2 is the slope of the altitude from A SOLVED EXAMPLES
then m1×m2 = −1.
∴ m2 = −1 = 2. Ex. 1) Find the acute angle between lines
m1 having slopes 3 and −2.
The slope of the altitude drawn from A is 2.
5.2.4 Angle between intersecting lines : Solution : Let m1 = 3 and m2 = −2.
We have obtained relation between slopes of
lines which are perpendicualr to each other. Let θ be the acute angle between them.
If given lines are not perpendicular to each
other then how to find angle between them? ∴ tanθ = m1 − m2 = 3 − (−2) = 5 =1
Let us derive formula to find the acute angle 1 + m1m2 1+ (3)(−2) −5
between intersecting lines.
∴ θ = 45°
Theorem : If θ is the acute angle between
non-vertical lines having slopes m1 and m2 then The acute angle between lines having slopes
3 and −2 is 45°.
tanθ = m1 − m2 Ex. 2) If the angle between two lines is 45°
1 + m1m2 and the slope of one of the lines is 1 , find
Proof : Let α and β be the inclinations of 2
non-vertical lines having slopes m1 and m2. the slope of the other line.
α ≠ 90° and β ≠ 90°.
Solution : If θ is the acute angle between lines
having slopes m1 and m2 then
tanθ = m1 − m2
1 + m1m2
Given θ = 45°.
Let m1 = 1. Let m2 be the slope of the other
line. 2
1 − m2 ∴ 1 = 1 − 2m2
2 2 + m2
Fig. 5.7 Fig. 5.8 tan 45° =
1
1 + 2 m2
∴ tanα = m1 and tanβ = m2 ∴ 1 − 2m2 = 1 or 1 − 2m2 = −1
From Fig. 5.7 and 5.8, we observe that
θ = β − α or θ = π − (β − α) 2 + m2 2 + m2
∴ tanθ = tan(β − α) or tanθ = tan{π−(β − α)} ∴ m2 = 3 or − 1
= −tan(β − α) 3
∴ tanθ = |tan(β − α)| = |tan(α − β)|
There are two lines which satisfy the given
∴ tatnaθnθ== tanα − tan β = m1 − m2 conditions.
1+ tanα tan β 1 + m1m2
Note that as θ is the acute angle, lines are not
perpendicular to each other. Hence m1 m2 ≠ −1.
∴ 1 + m1 m2 ≠ 0
108
EXERCISE 5.2 The y co-ordinate of every point on the X-axis
is 0 and this is true only for points on the
1. Find the slope of each of the following X-axis. Therefore, the equation of the X-axis
lines which passes through the points : is y = 0
The x co-ordiante of every point on the
(a) A(2,−1), B(4,3) (b) C(−2,3), D(5,7) Y-axis is 0 and this is true only for points
(c) E(2,3), F(2,−1) (d) G(7,1), H(−3,1) on the Y-axis. Therefore, the equation of the
Y-axis is x = 0 .
2. If the X and Y-intercepts of line L are 2 The equation of any line parallel to the
and 3 respectively then find the slope of Y-axis is of the type x = k (where k is a
line L. constant) and the equation of any line parallel
to the X-axis is of the type y = k. This is all
3. Find the slope of the line whose inclination about vertical and horizontal lines.
is 30°. Let us obtain equations of non-vertical
and non -horizontal lines in different forms:
4. Find the slope of the line whose inclination 5.3.1 Point-slope Form : To find the equation
is π . of the line having slope m and which passes
through the point A(x1, y1).
4
Fig. 5.9
5. A line makes intercepts 3 and 3 on the Proof : Let L be the line passing through the
co-ordiante axes. Find the inclination of point A(x1, y1) and which has slope m.
the line. Let P(x, y) be any point on the line L other
than A.
6. Without using Pythagoras theorem show Then the slope of line L = y − y1 .
that points A(4,4), B(3, 5) and C(−1, −1)
are the vertices of a right angled triangle. x − x1
But the slope of line L is m. (given)
7. Find the slope of the line which makes ∴ y − y1 = m
angle of 45° with the positive direction
of the Y-axis measured anticlockwise. x − x1
∴ The equation of the line L is
8. Find the value of k for which points
P(k,−1),Q(2,1) and R(4,5) are collinear. (y − y1) = m (x − x1)
9. Find the acute angle between the X-axis
and the line joining points A(3,−1) and
B(4,−2).
10. A line passes through points A(x1, y1) and
B(h, k). If the slope of the line is m then
show that k − y1 = m(h − x1).
11. If points A(h, 0), B(0, k) and C(a, b) lie
on a line then show that a + b = 1.
hk
5.3 Equation of line in standard forms : An
equation in x and y which is satisfied by the
co-ordinates of all points on a line and no
other points is called the equation of the line.
109
The equation of the line having slope m This is the equation of line L.
and passing through A(x1, y1) is The equation of line having slope m and which
(y − y1) = m (x − x1). makes intercept c on the Y-axis is y = mx + c.
Remark : In particular if the line passes Ex. Obtain the equation of line having slope
through the origin O(0,0) and has slope m, 3 and which makes intercept 4 on the Y-axis.
then its equation is y − 0 = m (x − 0)
Solution : The equation of line having slope
∴ y = mx m and which makes intercept c on the Y-axis
is
Ex. Find the equation of the line passing
through the point A(2, 1) and having slope −3. y = mx + c.
∴ the equation of the line giving slope 3
Soln. : Given line passes through the point and making Y-intercept 4 is y = 3x + 4.
A(2, 1) and slope of the line is −3.
5.3.3 Two-points Form : To find the equation
The equation of the line having slope m of line which passes through points A(x1, y1) and
and passing through A(x1, y1) is B(x2, y2).
(y − y1) = m (x − x1).
The equation of the required line is
y − 1 = −3(x − 2)
∴ y − 1 = −3x + 6
∴ 3x + y − 7 = 0
5.3.2 Slope-Intercept form : To find the
equation of line having slope m and which
makes intercept c on the Y-axis.
Fig. 5.11
Proof : A(x1, y1) and B(x2, y2) are the two
given points on the line L. Let P(x, y) be
any point on the line L, other than A and B.
Now points A and P lie on the line L.
The slope of line L = y − y1 .......... (1)
x − x1
Fig. 5.10
Proof : Let L be the line with slope m and Also points A and B lie on the line L.
which makes Y-intercept c. Line L meets the
Y-axis in the point C(0, c). ∴ Slope of line L = y2 − y1 .......... (2)
x2 − x1
Let P(x, y) be any point on the line other than From (1) and (2) we get y − y1 = y2 − y1
C. Then the slope of the line L is x − x1 x2 − x1
y−c = m (given) ∴ x − x1 = y − y1
x−0 x2 − x1 y2 − y1
∴ y−c =m This is the equation of line L.
x
As line is non-vertical and non-horizontal,
∴ y = mx + c x1 ≠ x2 and y1 ≠ y2.
110
The equation of the line which passes through ∴ The slope of the line L = Slope of AB
points A(x1, y1) and B(x2, y2) is x − x1 = y − y1 = b−0 = −b
x2 − x1 y2 − y1 0−a a
Ex. Obtain the equation of the line passing ∴ y = −b ∴ x + y =1
through points A(2, 1) and B(1, 2). x −a a ab
Solution : The equation of the line which passes This is the equation of line L.
through points A(x1, y1) and B(x2, y2) is
x − x1 = y − y1 . The equation of the line which makes
intercepts a and b on the co-ordiante axes
x2 − x1 y2 − y1 is x + y = 1 (a, b ≠ 0)
∴ The equation of the line passing through ab
points A(2, 1) and B(1, 2) is x − 2 = y −1 Ex. Obtain the equation of the line which
1− 2 2−1 makes intercepts 3 and 4 on the co-ordiante
axes.
∴ x − 2 = y −1 Solution : The equation of the line which
−1 1 makes intercepts a and b on the co-ordiante
∴ x − 2 = −y + 1 axes x + y = 1
∴x+y−3−0 ab
5.3.4 Double-Intercept form : To find the The equation of the line which makes intercepts
equation of the line which makes non-zero
intercepts a and b on the co-ordinate axes. 3 and 4 on the co-ordiante axes is x + y = 1
34
∴ 4x + 3y −12 = 0 .
Fig. 5.12 5.3.5 Normal Form : Let L be a line and
segment ON be the perpendicular (normal)
Let a be the X-intercept and b be the drawn from the origin to line L.
Y-intercept of line L. If ON = p and ray ON makes angle α with
the positive X −axis then to find the equation
∴ Line L meets the X-axis in A(a, 0) and the of line L.
Y-axis in B(0, b).
Fig. 5.13
Let P(x, y) be any point on line L other than
A and B.
∴ The slope of line L = slope of line AP
= y−0 = y
x−a x−a
111
From Fig. 5.13 we observe that N is ∴ 3 x+1 y=5
22
(p cosa, p sina)
Therefore slope of ON is p sinα − 0 = tanα ∴ 3x + y −10 = 0
pco sα − 0 Ex. 2) Reduce the equation 3x − y − 2 = 0
into normal form. Find the values of p and α .
As ON ierpendicular to line L,
Slope of line L = tanq = - cot a
And it passes through (p cosa, p sina) Solution : Comparing 3x − y − 2 = 0 with
ax + by + c = 0 we get a = 3 , b = −1 and
∴ By the point − slope form, the equation
c = −2 .
of line L is,
y − p sina = − cosα ( x - p cosα ) a2 + b2 = 3+1 = 2
sin α
Divide the given equation by 2.
y sin a - p sin2 a = x cos a + p cos2 a
x cos a + y sin a = (sin2 a cos2 a) 3 x− 1 y =1
2 2
x cos a + y sin a = p (p > o)
The equation of the line, the normal to which ∴ cos330° x=sin330° y = 1 is the required
from the origin has length p and the normal normal form of the given equation.
makes angle α with the positive directions of p = 1 and θ = 330ο .
the X-axis, is x cosα + y sinα = p.
SOLVED EXAMPLES Ex. 3) Find the equation of the line :
Ex. 1) The perpendicular drawn from the origin (i) parallel to the X -axis and 3 unit below it,
to a line has length 5 and the perpendicular (ii) passing through the origin and having
makes angle with the positive direction of the inclination 30°
X-axis. Find the equation of the line.
(iii) passing through the point A(5,2) and having
Solution : The perpendicular (normal) drawn slope 6
from the origin to a line has length 5.
∴ p=5 (iv) passing through the points A(2-1) and
The perpendicular (normal) makes angle 30° B(5,1)
with the positive direction of the X-axis. (v) having slope − 3 and y −intercept 5,
∴ θ = 30°
The equation of the required line is x cosa + 4
y sina = p (vi) making intercepts 3 and 6 on the co-ordinate
axes.
∴ x cos30° + y sin30° = p
(vii) passing through the point N ( −2,3) and
the segment of the line intercepted between
the co−ordinate axes is bisected at N.
112
Solution : (i) Equation of line parallel to the (vi) By using the double intercept form
X-axis is of the form : y = k , x + y =1
∴ the equation of the required line is ab
y=3
(ii) Equation of line through the origin and x −intercept = a = 3 ; y −intercept = b = 6 .
having slope m is of the form : y = mx . the equation of the required line is x + y = 1
36
2x + y −6 = 0
slope = m = tan = tan 300 = 3 (vii) Let the given line meet the X −axis
∴ the equation of the required line is in A(a,0) and the Y −axis in B(0,b).
y = 3 x ∴ 3x − y =0 The mid−point of AB is
(iii) By using the point−slope form d a + 0 , 0 + b = (−2, 3)
y-y1 = m (x-x1) 2 2
slope = m = -6
∴ a = - 2 and b =3
Equation of the required line is 2 2
(y-2) = -6 (x+5) ∴ a = - 4 b = 6
∴ By using the double intercept form :
6x + y + 28 = 0 x + y =1 x + y =1 ∴ 3x − 2 y +12 = 0
a b −4 6
(iv) By using the two points form
x − x1 = y − y1 An interesting property of a straight line.
x2 − x1 y2 − y1
Consider any straight line in a plane.
Here (x1, y1) = (2-1); (x2, y2) = (5,1) It makes two parts of the points which are
not on the line.
∴ the equation of the required line is
x − 2 = y +1
5− 2 1+1
∴ 2( x − 2) = 3( y +1)
∴ 2x −3y − 7 = 0
(v) By using the slope intercept form y = m x + c
Given m=−3, c=5 Fig. 5.14
4
Thus the plane is divided into 3 parts, the
∴ the equation of the required line is points on the line, points on one side of the
line and points on the other side of the line.
y = − 3 x + 5 ∴ 3x + 4 y − 20 = 0
4
113
If the line is given by ax+by+c = 0, then 4. Find the equation of the line
for all points (x1,y1) on one side of the line a) passing through the points A(2,0) and
ax1+by1+c > 0 and for all points (x2,y2) on the
other side of the line, ax2+by2+c < 0. B(3,4).
For example, consider the line given by b) passing through the points P(2,1) and
Q(2,-1)
y − 2x − 3 = 0. Points P(−2,0), Q(−2,4),
5. Find the equation of the line
R( 1 ,5) lie on one side and at each of those
2 a) containing the origin and having
inclination 60°.
points, y − 2x − 3 > 0. The points A(0,0),
b) passing through the origin and parallel
B( 1 ,3), C(8,4) lie on the other side of the line to AB, where A is (2,4) and B is (1,7).
2
c) having slope 1 and containing the
and at each of those points y − 2x − 3 < 0. point (3,−2). 2
Activity : d) containing the point A(3,5) and having
slope 2 .
Draw the straight lines given by 2y + x 3
= 5, x = 1, 6y − x + 1 = 0 give 4 points on
each side of the lines and check the property e) containing the point A(4,3) and having
stated above. inclination 120°.
EXERCISE 5.3 f) passing through the origin and which
bisects the portion of the line 3x + y = 6
1. Write the equation of the line : intercepted between the co−ordinate
a) parallel to the X−axis and at a distance axes.
of 5 unit form it and above it. 6. Line y = mx + c passes through points
A(2,1) and B(3,2). Determine m and c.
b) parallel to the Y− axis and at a distance
of 5 unit form it and to the left of it. 7. Find the equation of the line having
inclination 135° and making X-intercept 7.
c) parallel to the X− axis and at a distance
of 4 unit form the point (−2, 3). 8. The vertices of a triangle are A(3,4), B(2,0)
and C(−1,6). Find the equations of the lines
2. Obtain the equation of the line : containing
a) parallel to the X−axis and making an (a) side BC (b) the median AD
intercept of 3 unit on the Y−axis. (c) the mid points of sides AB and BC.
b) parallel to the Y−axis and making an
9. Find the x and y intercepts of the
intercept of 4 unit on the X−axis.
following lines :
3. Obtain the equation of the line containing
the point : (a) x + y = 1 (b) 3x + 2 y = 1
32 23
a) A(2,−3) and parallel to the Y−axis.
(c) 2x − 3y +12 = 0
b) B(4,-3) and parallel to the X−axis.
114
10. Find equations of lines which contains Remark : If a=0 then the line is parallel to
the point A(1,3) and the sum of whose the X-axis. It does not make intercept on the
intercepts on the co−ordinate axes is zero. X-axis.
11. Find equations of lines containing the If b=0 then the line is parallel to the Y-axis.
point A(3,4) and making equal intercepts on It does not make intercept on the Y-axis.
the co−ordinates axes.
SOLVED EXAMPLES
12. Find equations of altitudes of the
triangle whose vertices are A(2,5), B(6,-1) Ex. 1) Find the slope and intercepts made by
and C(-4,-3). the following lines :
(a) x + y +10 = 0 (b) 2x + y + 30 = 0
13. Find the equations of perpendicular (c) x + 3y −15= 0
bisectors of sides of the triangle whose Solution : (a) Comparing equation x+y+10=0
vertices are P(−1,8), Q(4,−2) and R(−5,−3). with ax + by + c = 0 ,
14. Find the co-ordinates of the orthocenter of we get =a 1,=b 1, c = 10
the triangle whose vertices are A(2,−2),
B(1,1) and C(−1,0). ∴ Slope of this line = − a = −1
b
15. N(3,−4) is the foot of the perpendicular
drawn from the origin to line L. Find the The X–intercept is −c = − 10 = −10
equation of line L. a 1
5.4 General form of equation of a line: We The Y-intercept is =− c = − 10 = −10
can write equation of every line in the form b 1
ax+by+c=0
(b) Comparing equation 2x + y + 30 = 0
This form of equation of a line is called the with ax + by + c = 0 .
general form.
The general form of y = 3x + 2 is 3x − y + 2 = 0 we get =a 2=, b 1 , c =30
x y ∴ Slope of this line = −a = −2
2 3 b
The general form of + =1 is 3x + 2 y − 6 = 0
−c − 30
The slope of the line ax + by + c = 0 is The X–intercept is a = 2 = −15
− a if b≠0 . The Y-intercept is −c = − 30 = −30
b b 1
The X–intercept is − c if a ≠ 0 . (c) Comparing equation x + 3y −15 = 0
a with ax + by + c = 0 .
The Y-intercept is − c if b≠0 . we get a =1, b =3 , c = −15
b
∴ Slope of this line = −a = −1
b 3
115
The x – intercept is −c = − −15 = 15 Solution : Slopes of the given lines are
a 1
m1 = 3 and m2 = 1.
3
The y – intercept is −c = − −15 = 5
b 3
The acute angle θ between lines having
Ex. 2) Find the acute angle between the slopes m1 and m2 is given by
following pairs of lines :
a) 12x −4 y =5 and 4x + 2 y =7 tanθ = m1 − m2
b) y = 2x + 3 and y = 3x + 7 1+ m1m2
Solution : (a) Slopes of lines 12x −4 y =5 and tanθ = 1 = 3− 1 3− 1 3− 1
3 3 3= 3
3× 1 =
1+1 2
4x + 2 y =7 are m1 = 3 and m2 = 2 . 1+
If θ is the acute angle between lines having 3
slope m1 and m2 then
= 1− 3 = −1 = 1
tanθ = m1 − m2 23 3 3
1+ m1m2
∴ tanθ = 1 ∴ θ = 300
3
∴ tanθ = 3− (−2) = 5 = 1 Ex. 4) Show that following pairs of lines are
1+ (3)(−2) −5 perpendicular to each other.
a) 2x −4 y = 5 and 2x + y = 17 .
∴ tanθ =1 ∴ θ =45°
b) y = 2x + 23 and 2x + 4 y = 27
(b) Slopes of lines y = 2x + 3 and y = 3x + 7
are m1 = 2 and m2 = 3 Solution : (i) Slopes of lines 2x −4 y =5 and
The acute angle θ between lines having
slopes m1 and m2 is given by 2x + y =17 are m1 = 1 and m2 = −2
2
tanθ = m1 − m2
1+ m1m2 Since m1.m 2 = 1 ×(−2) = −1, given lines
2
are perpendicular to each other.
∴ tanθ = 2−3 = −1 = 1 (ii) Slopes of lines y = 2x + 23 and
7 7
1+ (2)(3) 2x+4y = 27 are m1 = − 1 and m2 = 2.
2
1
∴ θ = ta n −1 7 . − 1×(2)
Since m1.m2 = 2 = −1, given lines
Ex. 3) Find the acute angle between the lines are perpendicular to each other.
y − 3x +1 = 0 and 3y − x + 7 = 0 .
116
Ex. 5) Find equations of lines which pass ∴ It’s equation is given by the formula
through the origin and make an angle of 45° ( y − y1 ) = m( x − x1 )
with the line 3x - y = 6. ∴ ( y − 7) = − 1 ( x −2)
Solution : Slope of the line 3x - y = 6. is 3. 3
Let m be the slope of one of the required ∴ 3y − 21= x + 2
lines. The angle between these lines is 45° . ∴ x + 3y = 23 .
∴ tan450 = m−3 Ex. 8) A line is perpendicular to the line
3x + 2 y −1 = 0 and passes through the point
1+ (m)(3)
A(1,1) . Find its equation.
m−3
∴1= 1+ 3m ∴ 1+ 3m = m –3 Solution : Slope of the line 3x + 2 y −1 is − 3 .
2
∴ 1+ 3m = m − 3 or 1+ 3m = − (m − 3) Required line is perpendicular it.
∴ m = −2 or 1 The slope of the required line is 2 .
2 3
Slopes of required lines are m1 = -2 and Required line passes through the point
1 A (1,1) .
m2 = 2
∴ It’s equation is given by the formula
Required lines pass through the origin. ( y − y1 ) = m( x − x1 )
∴ Their equations are y = −2x and
∴ ( y − 1) = 2 ( x −1)
3
1
y = 2 x ∴ 3y −3= 2x − 2
∴ 2x −3y +1= 0.
∴ 2x + y = 0 and x − 2 y = 0 Note:
Ex. 6) A line is parallel to the line 2x + y = 7 Point of intersection of lines :
and passes through the origin. Find its equation.
The co-ordinates of the point of intersection
Solution : Slope of the line 2x + y = 7 is −2. of two intersecting lines can be obtained by
Required line passes through the origin. solving their equations simultaneously.
∴ It’s equation is y = −2x Ex. 9) Find the co-ordinates of the point of
∴ 2x + y = 0. intersection of lines x + 2 y = 3 and 2x − y =1.
Ex. 7) A line is parallel to the line x + 3y = 9 Solution : Solving equations x + 2 y = 3 and
and passes through the point A(2,7). Find its 2x − y = 1 simultaneously, we get x = 1 and
equation.
y = 1.
Solution : Slope of the line x + 3y = 9 is − 1 ∴ Given lines intersect in point (1,1) .
3
Required line passes through the point
A(2,7).
117
Ex. 10) Find the equation of line which is AB2 = c 2 + c 2 = c2 a2 + b2
parallel to the X-axis and which passes through a b a 2b2
the point of intersection of lines x + 2 y = 6
and 2x − y = 2
∴ AB = c a2 + b2
Solution : Solving equations x + 2 y = 6 and ab
2x − y = 2 simultaneously, we get x = 2
and y =2 . Now,
∴ The required line passes through the Area of ∆OAB = 1 AB× ON
2
point (2, 2) .
1c a 2 + b2 × p ... (I)
As it is parallel to the X-axis, its equation = 2 ab
is y = 2.
But, Area of ∆OAB = 1 OA × OB
2
5.4.1 The distance of the Origin from a 1 c c c2
Line : = 2 a b = 2ab ... (II)
The distance of the origin from the line From (I) and (II), we get
ax + by + c = 0 is given by p = c p= c
a2 + b2 a2 + b2
5.4.2 The distance of the point (x1,y1) from a
line: The distance of the point P (x1,y1) from
line ax + by + c = 0 is given by
p = ax1 + by1 + c
a2 + b2
Fig. 5.15
Proof : Let A and B be the points where line
ax + by + c = 0 cuts the co-ordinate axes.
∴ A − c , 0 and B 0, − c
a b
cc Fig. 5.16
OA = a and OB = b
By Pythagoras theorem AB2 = OA2 +OB2
118
Proof : If line ax + by + c = 0 cuts co-ordiante
axes in B and C respectively then B is − c , 0
a
and C is 0, − c .
b
Let PM be perpendicular to ax + by + c = 0 .
Let PM = p
A ( ∆PBC ) = 1 BC × PM = 1 p c2 c2 = pc a2 + b2 Fig. 5.17
2 2 a2 + b2 2ab
.(I) Proof : To find the distance between parallel
lines, we take any one point on any one of
Now P ( x1, y1 ) , B − c , 0 and C 0, − c these two lines and find its distance from the
a b other line.
are vertices of PBC. A − c1 , 0 is a point on the first line.
a
∴ 1 1 1 . .(II)
A (∆PBC) = 1 x1 − c 0 = 1 x1 c + y1 c + c2 Its distance from the line ax + by + c2 = 0
2 a 2 b a ab is given by
y1 0 − c
b
c1
a − a + b ( 0) + c2 −c1 + c2 c1 − c2
a2 + b2
From (I) and (II) we get p= a2 + b2 = a2 + b2 =
pc a 2 + b2 = 1 x1 c + y1 c + c2
2ab 2 b a ab
SOLVED EXAMPLES
∴ pc a2 + b2 =| acx1 + bcy1 + c2 | Ex. 1) Find the distance of the origin from
∴ p a2 + b2 =| ax1 + by1 + c | the line 3x + 4 y +15= 0
∴ p = ax1 + by1 + c Solution : The distance of the origin from the
a2 + b2 line ax + by + c = 0 is given by
5.4.3 The distance between two parallel p= c
lines : a2 + b2
Theorem : The distance between the parallel
lines ax + by + c1 = 0 and ax + by + c2 = 0 is give ∴ The distance of the origin from the line
3x + 4 y +15= 0 is given by
by p = c1 − c2
a2 + b2 p= 15 = 15 =3
32 + 42 5
119
Ex. 2) Find the distance of the point P (2,5) lines is of the form ( y + 3)= m( x −2) . This set
from the line 3x + 4 y +14 = 0 is also form a family of lines.
Consider the set of all lines which are
Solution : The distance of the point P ( x1, y1 ) parallel to the line y = x . They all have the
same slope.
from the line ax + by + c = 0 is given by
The set of all lines which pass through
p = ax1 + by1 + c a fixed point or which are parallel to each
a2 + b2 other is a family of lines.
Interpretation of u + kv = 0 : Let
∴ The distance of the point P (2,5) from
u ≡ a1x + b1 y + c1 and v ≡ a2 x + b2 y + c2
the line 3x + 4 y +14 = 0 is given by
Equations u=0 and v=0 represent two lines.
p = 3(2) + 4 (5) +14 = 40 = 8 Equation u + kv = 0, k ∈ R represents a family
32 + 42 5 of lines.
Ex. 3) Find the distance between the parallel Fig. 5.18
lines 6x +8y + 21= 0 and 3x + 4 y +7 = 0 .
Solution : We write equation 3x + 4 y + 7 = 0 as
6x +8y +14 = 0 in order to make the
coefficients of x and coefficients of y in
both equations to be same.
Now by using formula p = c1 − c2
a2 + b2
We get the distance between the given
parallel lines as
p = 21−14 = 7 In equation u + kv = 0 let
62 + 82 10
u ≡ a1x + b1y + c1 , v ≡ a2 x + b2 y + c2
5.4.4 Family of Lines : A set of lines which
have a common property is called a family We get (a1x + b1 y + c1 ) + k (a2 x + b2 y + c2 ) = 0
of lines. Consider the set of all lines passing (a1 + ka2 ) x + (b1 + k b2 ) y + (c1 + k c2 ) = 0
through the origin. Equation of each of these
lines is of the form y = mx . This set of lines Which is a first degree equation in x and y.
is a family of lines. Different values of m give Hence it represents a straight line.
different lines.
i) If lines u = 0 and v = 0 intersect each
Consider the set of all lines which pass through ( )other in P x1, y1 then a1x1 + b1y1 + c1 = 0 and
the point A (2, −3) . Equation of each of these a2 x1 + b2 y1 + c2 = 0
120
Therefore ∴ 7k = 16 ∴ 16
k= 7
(a1x1 + b1 y1 + c1 ) + k (a2 x1 + b2 y1 + c2 ) = 0 + k0 = 0
∴ the equation of the required line is
Thus line u + kv = 0 passes through the point
∴ ( x + 2 y + 6) + 16 (2x − y −2) = 0
of intersection of lines u = 0 and v = 0 for 7
every real value of k.
∴ (7x +14 y + 42) + (32x −16 y − 32) = 0
ii) If lines u = 0 and v = 0 are parallel to ∴ 39x − 2 y +10 = 0
each other then their slope is same.
∴ − a1 = − a2 Ex. 2) Find the equation of line which passes
b1 b2 through the point of intersection of lines
3x + 2 y − 6 = 0 , x + y +1 = 0 and the point
∴ each ratio = − a1 + ka2
b1 + kb2 A (2,1) .
= slope of the line u+kv=0 Solution : Since the required line passes
through the point of intersection of lines
∴ Slopes of lines =u 0=, v 0 and
3x + 2 y − 6 = 0 and x + y +1 = 0 , its
u + kv = 0 are the same. equation is of the form u + k v = 0 .
∴ Line u + kv = 0 is parallel to lines
=u 0=,v 0 .
SOLVED EXAMPLES ∴ (3x + 2 y − 6) + k ( x + y +1) = 0
Ex. 1) Find the equation of the line which ∴ (3 + k)x + (2 + k ) y + (−6 + k ) = 0
passes through the point of intersection of
lines x + 2 y + 6 = 0 , 2x − y = 2 and which This line passes through the point A(2,1) .
makes intercept 5 on the Y-axis.
∴ (2, 1) satisfy this equation.
Solution : As the required line passes through ∴ (3 + k)(2) + (2 + k )(1) + (−6 + k ) = 0
the point of intersection of lines x + 2 y + 6 = 0 ∴ 4k + 2 = 0 ∴ k=−1
and 2x − y = 2 , its equation is of the form 2
u + k v = 0 . ∴ The equation of the required line is
∴ (x + 2y +6)+ k (2x − y −2) = 0 (3x + 2 y − 6) + − 1 ( x + y +1) = 0
2
∴ (1+ 2k ) x + (2 −1k ) y + (6 − 2k ) = 0 5x + 3y −13 = 0
The Y-intercept of this line is given 5.
∴ − 6 − 2k = 5 ∴ −6 + 2k = 10 5 k
2−k
121
EXERCISE 5.4 10) Find the equation of the line whose
X-intercept is 3 and which is perpendicular
1) Find the slope, X-intercept, Y-intercept to the line 3x − y + 23 =0 .
of each of the following lines.
11) Find the distance of the origin from the line
a) 2x + 3y − 6 =0 b) 3x−y−9=0 7x +24y-50 = 0 .
c) x + 2 y =0
12) Find the distance of the point A (−2,3) from
2) Write each of the following equations in the line 12x − 5y −13 =0.
ax + by + c =0 form.
13) Find the distance between parallel lines
a) y = 2x-4 b) y = 4 4x − 3y + 5 =0 and 4x − 3y + 7 = 0
c) x + y =1 d) x - y = 0 14) Find the distance between parallel lines
24 32 9x + 6 y − 7 =0. and 3x + 2y + 6 = 0
3) Show that lines x − 2 y − 7 =0 and 15) Find points on the line x + y − 4 =0 which
2x − 4y + 15 = 0 are parallel to each other. are at one unit distance from the line
x + y − 2 =0
4) Show that lines x − 2 y − 7 =0 and
2x + y + 1 = 0 are perpendicular to each 16) Find the equation of the line parallel to the
other. Find their point of intersection. X-axis and passing through the point of
intersection of lines x + y − 2 =0 and
5) If the line 3x + 4 y =p makes a triangle 4x + 3y = 10.
of area 24 square unit with the co-ordinate
axes then find the value of p . 17) Find the equation of the line passing
through the point of intersection of lines
6) Find the co-ordinates of the foot of the x + y − 2 =0 and 2x − 3y + 4 =0 and
perpendicular drawn from the point A(-2,3) making intercept 3 on the X-axis.
to the line 3x − y −1 =0 .
7) Find the co-ordinates of the circumcenter 18) If A(4,3), B(0,0), and C(2,3) are the
of the triangle whose vertices are vertices of ∆ ABC then find the equation of
A(−2,3), B(6, −1),C(4,3). bisector of angle BAC.
8) Find the co-ordinates of the orthocenter of 19) D(-1,8), E(4,-2), F(−5,-3) are midpoints of
the triangle whose vertices are A(3,-2),
B(7,6), C (-1,2) sides BC, CA and AB of ∆ ABC . Find
(i) equations of sides of ∆ ABC .
9) Show that lines
(ii) co-ordinates of the circumcenter of
3x − 4 y +=5 0,7x − 8 y +=5 0,and 4x + 5y − 4=5 0 ∆ ABC .
are concurrent. Find their point of 20) O(0, 0), A(6,0) and B(0,8) are vertices of a
concurrence. triangle. Find the co-ordinates of the incenter
of ∆OAB.
122
Let's Remember • Angle between intersecting lines : If θ is
the acute angle between lines having slopes
• Locus : A set of points in a plane which m1 and m2 then tanθ = m1 − m2
satisfy certain geometrical condition (or 1+ m1m2
conditions) is called a locus.
• Equations of line in different forms :
• Equation of Locus : Every point in
XY plane has Cartesian co-ordinates. An • Slope point form : ( y − y1)= m ( x − x1 )
equation which is satisfied by co-ordinates
of all points on the locus and which is not • Slope intercept form : y = mx + c
satisfied by the co-ordinates of any point
which does not lie on the locus is called • Two points form : x − x1 = y − y1
the equation of the locus. x2 − x1 y2 − y1
• Inclination of a line : The smallest angle • Double intercept form : x + y = 1
θ made by a line with the positive direction ab
of the X-axis, measured in anticlockwise
sense, is called the inclination of the line. • Normal form : x cosα + y sinα = p
Clearly 0° ≤ θ ≤ 180°. • General form : ax + by + c = 0
• Slope of a line : If θ is the inclination of • Distance of a point from a line :
a line then tanθ (if it exist) is called the • The distance of the origin from the line
slope of the line. ax + by + c = 0 is given by p = c
a2 + b2
If A( x1, y1 ) , B ( x2 , y2 ) be any two points • The distance of the point P ( x1, y1 ) from line
on the line whose inclination is θ then ax + by + c = 0 is given by
tanθ = y2 − y1 (if x1 ≠ x2) p = ax1 + by1 + c
x2 − x1 a2 + b2
• Perpendicular and parallel lines : Lines • The distance between the Parallel lines
having slopes m1 and m2 are perpendicular : The distance between the parallel lines
to each other if and only if m1m2 = −1. ax + by + c1 = 0 and ax + by + c2 = 0 is
give by p = c1 − c2
Two lines are parallel if and only if they a2 + b2
have the same slope.
123
MISCELLANEOUS EXERCISE - 5 (A) 3x ±y+6=0 (B) 3x +y±6=0
(C) x+y=6 (D) x+y=−6
(I) Select the correct option from the given
alternatives. 7) A line passes through (2,2) and is
perpendicular to the line 3x+y=3. Its
1) If A is (5,−3) and B is a point on the y−interecpt is
x−axis such that the slope of line AB is
−2 then B ≡ (A) 1 (B) 2 (C) 1 (D) 4
3 3 3
(A) (7,2) (B) ( 7 ,0)
2
(C) (0, 7 ) (D) ( 2 ,0) 8) The angle between the line 3x −y−2=0
2 7 and x− 3y +1=0 is
2) If the point (1,1) lies on the line passing (A) 15° (B) 30° (C) 45° (D) 60°
through the points (a,0) and (0,b), then
9) If kx+2y−1=0 and 6x−4y+2=0 are identical
1+1 = lines, then determine k.
ab
(A) −1 (B) 0 (C) 1 1 (A) −3 (B) −1 (C) 1 (D) 3
(D) ab 3 3
3) If A(1,−2), B(−2,3) and C(2,−5) are the 10) Distance between the two parallel lines
vertices of ∆ABC, then the equation of y=2x+7 and y=2x+5 is
the median BE is
(A) 2 (B) 1 (C) 5 (D) 2
(A) 7x+13y+47=0 (B) 13x+7y+5=0 5 5 2 5
(C) 7x−13y+5 =0 (D) 13x−7y−5=0 (II) Answer the following questions.
1) Find the value of k
4) The equation of the line through (1,2),
which makes equal intercepts on the axes, a) if the slope of the line passing through
is
the points P(3, 4), Q(5, k) is 9.
(A) x+y=1 (B) x+y=2 b) the points A (1,3), B(4, 1), C(3, k) are
(C) x+y=4 (D) x+y=3 collinear
c) the point P(1,k) lies on the line passing
5) If the line kx+4y=6 passes through the
point of intersection of the two lines through the points A(2, 2) and B(3, 3).
2x+3y=4 and 3x+4y=5, then k =
2) Reduce the equation 6x + 3y + 8 = 0 into
(A) 1 (B) 2 (C) 3 (D) 4 slope-intercept form. Hence find its slope.
6) The equation of a line, having inclination 3) Find the distance of the origin from the
120° with positive direction of X−axis, line x=−2.
which is at a distance of 3 units from
the origin is 4) Does point A (2,3) lie on the line
3x + 2 y − 6 = 0 ? Give reason.
124
5) Which of the following lines passes 12) Find the equation of the line which
through the origin ? contains the point A(3,5) and makes equal
intercepts on the co−ordinates axes.
(a) x = 2 (b) y = 3
(c) y = x + 2 (d) 2x − y = 0 13) The vertices of a triangle are A(1,4),
B(2,3) and C(1,6). Find equations of
6) Obtain the equation of the line which is :
(a) the sides (b) the medians
a) parallel to the X−axis and 3 unit below
it. (c) Perpendicular bisectors of sides
b) parallel to the Y−axis and 2 unit to the (d) altitudes of ∆ ABC.
left of it.
14) Find the equation of the line which passes
c) parallel to the X−axis and making an through the point of intersection of lines
intercept of 5 on the Y−axis. x + y − 3 = 0 , 2x − y +1 = 0 and which is
parallel X- axis.
d) parallel to the Y−axis and making an
intercept of 3 on the X−axis. 15) Find the equation of the line which passes
through the point of intersection of lines
7) Obtain the equation of the line containing x + y + 9 = 0 , 2x + 3y +1 = 0 and which
the point makes X-intercept 1.
(i) (2,3) and parallel to the X−axis. 16) Find the equation of the line through A
(ii) (2,4) and perpendicular to the Y−axis.
(−2,3) and perpendicular to the line
8) Find the equation of the line : through S (1, 2) and T(2,5) .
a) having slope 5 and containing point 17) Find the X-intercept of the line whose
slope is 3 and which makes intercept 4
A (−1, 2) . on the Y−axis.
b) containing the point T(7,3) and having 18) Find the distance of P(−1,1) from the line
inclination 900 . 12( x + 6) = 5( y − 2).
c) through the origin which bisects 19) Line through A(h,3) and B (4,1) intersect
the portion of the line 3x + 2 y = 2
intercepted between the co−ordinate the line 7x − 9 y −19 = 0 at right angle.
axes. Find the value of h.
9) Find the equation of the line passing 20) Two lines passing through M (2,3) intersect
through the points S(2,1)and T(2,3)
each other at an angle of 45°. If slope
10) Find the distance of the origin from the of one line is 2, find the equation of the
line 12x + 5y + 78= 0 other line.
11) Find the distance between the parallel 21) Find the Y-intercept of the line whose
lines 3x +4 y + 3= 0 and 3x +4 y +15= 0 slope is 4 and which has X intercept 5.
125
22) Find the equations of the diagonals of the 27) Find points on the X-axis whose distance
rectangle whose sides are contained in the from the line x + y = 1 is 4 unit.
lines=x 8=, x 10, y = 11 and y = 12 . 34
23) A(1, 4), B(2,3) and C (1, 6) are vertices 28) The perpendicular from the origin to a line
of ∆ABC. Find the equation of the altitude
through B and hence find the co-ordinates meets it at (−2,9). Find the equation of
of the point where this altitude cuts the
side AC of ∆ABC. the line.
24) The vertices of ∆ PQR are P(2,1), 29) P (a,b) is the mid point of a line segment
Q(-2,3) and R(4,5). Find the equation of between axes. Show that the equation of
the median through R. the line is x + y = 2 .
25) A line perpendicular to segment joining A ab
(1,0) and B (2,3) divides it internally in 30) Find the distance of the line 4x − y = 0
the ration 1:2. Find the equation of the from the point P (4,1) measured along
line.
the line making an angle of 1350 with the
26) Find the co-ordinates of the foot of the positive X-axis.
perpendicular drawn from the point P
31) Show that there are two lines which pass
(−1,3) the line 3x-4y-16 = 0 . through A(3,4) and the sum of whose
intercepts is zero.
32) Show that there is only one line which
passes through B(5,5) and the sum of
whose intercept is zero.
126
6 Circle
Let's Study \ x2 + y2 = r2.
This is the standard equation of a circle.
Equation of a circle and its different forms
Equation of Tangent to a circle (2) Centre-radius form :
Condition for tangency In Fig. 6.2, C(h, k)
Director circle is the centre and r
is the radius of the
Let's Recall circle. P(x, y) is any
point on the circle.
Properties of chords and tangents of a circle.
The angle inscribed in a semicircle is a right ∴ CP = r
angle. Also,
Product of slopes of perpendicular lines is -1. Fig. 6.2
Slopes of parallel lines are equal.
CP = ( x − h)2 + ( y − k )2 \ r2 = (x - h)2 + (y - k)2
A circle is a set of all points in a plane (x − h)2 + (y − k)2 = r2 is the centre-radius form of
which are equidistant from a fixed point in the equation of a circle.
plane.
(3) Diameter Form : In the Fig. 6.3, C is the
centre of the circle.
The fixed point is called the centre of the A(x1 , y1), B(x2 , y2)
circle and the distance from the centre to any
point on the circle is called the radius of the circle. are the end points
of a diameter of the
circle. P(x, y) is any
6.1 Different forms of equation of a circle point on the circle.
(1) Standard form : In Fig. 6.1, the origin, O Angle inscribed in a
is the centre of the semi circle is a right Fig. 6.3
circle. P(x, y) is any angle; hence, m∠ APB = 90ο; that is AP ⊥ BP.
point on the circle. y − y1 y − y2
x − x1 x − x2
The radius of circle Slope of AP = and slope of BP = ,
is r. As AP ⊥ BP, product of their slopes is –1.
∴ OP = r. ∴ y − y1 × y − y2 = –1
x − x1 x − x2
By distance formula
OP2 = ( x − 0)2 + ( y − 0)2 (y – y1) (y – y2) = –(x – x1)(x – x2)
∴ we get r2 = x2 + y2
Fig. 6.1 (x – x1) (x – x2) + (y – y1)(y – y2) = 0
127
That is, (x - x1)(x - x2) + (y - y1) (y - y2) = 0 Ex.4 Find the equation of circle touching the
Y-axis at point (0, 3) and whose Centre is
This is called the diameter form of the equation at (–3, 3).
of circle, where (x1, y1) and (x2, y2) are endpoints
of diameter of the circle. Solution :
The circle touches
SOLVED EXAMPLES the Y-axis at point
C(−3, 3) (0,3) (0,3), and the centre
Ex.1 Find the equation of a circle with centre at is (−3,3) we get
origin and radius 3. radius r = 3
Solution : Standard equation of a circle is
x2 + y2 = r2 here r = 3 Fig. 6.4 By using centre
radius form;
∴ x2 + y2 = 32
x2 + y2 = 9 is the equation of circle (x − h)2 + (y − k)2 = r2
Ex.2 Find the equation of a circle whose centre is (x + 3)2 + (y – 3)2 = 9
(−3, 1) and which pass through the point (5, 2).
x2 + 6x + 9 + y2 − 6y + 9 = 9
Solution : Centre C = (−3, 1), ∴ x2 + y2 + 6x – 6y + 9 = 0 is the equation
Circle passes through the point P(5, 2). of the circle.
By distance formula,
Ex.5 Find the equation of the circle whose centre
r2 = CP2 = (5 + 3)2 + (2 − 1)2 is at (3,−4) and the line 3x−4y−5 = 0 cuts the circle
at A and B; where l(AB) = 6.
= 82 + 12 = 64 + 1 = 65
∴ the equation of the circle is Solution : centre of the
circle C(h,k) = C(3,−4)
(x + 3)2 + (y − 1)2 = 65 (centre-radius form)
x2 + 6x + 9 + y2 − 2y + 1 = 65 3x − 4y − 5 = 0 cuts the
circle at A and B.
x2 + y2 + 6x − 2y + 10 − 65 = 0
x2 + y2 + 6x − 2y − 55 = 0 l(AB) = 6
Ex.3 Find the equation of the circle with A(2, −3) CM ⊥ AB
and B(−3, 5) as end points of its diameter.
Fig. 6.5 ∴ AM = BM = 3
Solution : By using the diameter form;
A (2, –3) ≡ (x1, y1) and B(–3, 5) ≡ (x2, y2) are CM = Length of perpendicular from centre on
the co-ordinates of the end points of a diameter of the line
the circle.
3(3) − 4(−4) − 5
∴ by the diameter form, equation of the circle is =
(3)2 + (−4)2
(x − x1) (x − x2) + (y − y1) (y − y2) = 0 = 9 +16 − 5
∴ (x – 2) (x + 3) + (y + 3) (y – 5) = 0 9 +16
∴ x2 + x – 6 + y2 – 2y – 15 = 0
20
∴ x2 + y2 + x – 2y – 21 = 0 = 5 = 4
128
From right angled triangle AMC 5. If y = 2x is a chord of circle x2 + y2 −10x = 0,
CA2 = CM2 + AM2 find the equation of circle with this chord as
= (4)2 + (3)2 diametre.
= 16 + 9 = 25
CA = radius of the circle = 5 6. Find the equation of a circle with radius 4
By centre radius form equation of the circle. units and touching both the co-ordinate axes
(x −3)2 + (y+4)2 = 52 having centre in third quadrant.
x2−6x+9+y2+8y+16 = 25
x2+y2−6x+8y = 0 7) Find the equation of circle (a) passing through
the origin and having intercepts 4 and −5 on
EXERCISE 6.1 the co-ordinate axes.
1. Find the equation of the circle with 8) Find the equation of a circle passing through
(i) Centre at origin and radius 4. the points (1,−4), (5,2) and having its centre
(ii) Centre at (−3, −2) and radius 6. on the line x−2y+9 = 0 .
(iii) Centre at (2, −3) and radius 5.
(iv) Centre at (−3, −3) passing through point Activity :
(−3, −6) (1) Construct a circle in fourth quadrant having
2. Find the centre and radius of the circle. radius 3 and touching Y-axis. How many
such circles can be drawn?
(i) x2 + y2 = 25 (ii) (x − 5)2 + (y − 3)2 = 20
(2) Construct a circle whose equation is
(iii) x − 1 2 + y + 1 2 = 1 x2 + y2 − 4x + 6y − 12 = 0. Find the area of
2 3 36
the circle.
3. Find the equation of the circle with centre
(i) At (a, b) touching the Y-axis 6.2 General equation of a circle :
(ii) At (-2, 3) touching the X- axis The general equation of a circle is of the
(iii) on the X-axis and passing through the
form x2 + y2 + 2gx + 2fy + c = 0, if g2 + f 2−c>0
origin having radius 4. The centre-radius form of equation of a circle
(iv) at (3,1) and touching the line
is
8x − 15y + 25 = 0 (x − h)2 + (y − k)2 = r2
i.e x2 − 2hx + h2 + y2 − 2ky + k2 = r2
4. Find the equation circle if the equations of i.e x2 + y2 − 2hx − 2ky + (h2 + k2 − r2) = 0
two diameters are 2x + y = 6 and 3x + 2y = 4. If this is the same as equation x2 + y2 + 2gx +
When radius of circle is 9. 2fy + c = 0, then comparing the coefficients
2g = – 2h, 2f = – 2k and c = h2 + k2 – r2.
∴ (h, k) ≡ (–g, –f) is the center and
r2 = h2 + k2 – c i.e. r = g 2 +f 2 − c is the radius.
129
Thus, Lets Note :
the general equation of a circle is
x2 + y2 + 2g x + 2 fy + c = 0 whose centre is The general equation of a circle is a second degree
(–g, –f) and radius is g 2 +f 2 − c . equation in x and y, coefficient of xy is zero,
coefficient of x2 = coefficient of y2
Activity : SOLVED EXAMPLES
Consider the general equation of a circle
x2 + y2 + 2gx + 2 fy + c = 0 Ex. 1) Prove that 3x2 + 3y2 − 6x + 4y − 1 = 0,
x2 + 2gx + + y2 + 2 fy + = + – c
∴ (x + )2 + (y + )2 = g2 + f 2 – c. represents a circle. Find its centre and radius.
∴ [x – ( )]2 + [y – ( )]2 = ( )2
(use centre radius form of equation of the circle). Solution : Given equation is
Therefore centre of the circle is (–g, –f ) and radius
is g 2 +f 2 − c . 3x2 + 3y2 − 6x + 4y − 1 = 0
dividing by 3, we get
x2 + y2 − 2x + 4y − 1 = 0 comparing
3 3
this with
x2 + y2 + 2gx + 2fy + c = 0
we get 2g = −2 ∴ g = –1
Let's Remember 2f = 4 ∴f= 2 and c = −1
3 3 3
(1) If g2 + f 2 − c > 0, the equation
x2 + y2 + 2gx + 2fy + c = 0 represents a circle g2 + f 2 − c = (−1)2 + 2 2 − −1
3 3
in the xy plane.
=1+ 4 + 1 = 16
(2) If g2 + f 2 − c = 0, then the equation 9 3 9
x2 + y2 + 2gx + 2fy + c = 0 represents a point
16 g2 + f2 − c > 0
which is true degenerate conic and is the
limiting position (radius is 0). As 9 > 0
(3) If g2 + f 2 − c < 0, then the equation ∴ 3x2 + 3y2 − 6x + 4y − 1 = 0 represents a
x2 + y2 + 2gx + 2fy + c = 0 does not represent circle.
∴ Centre of the circle = (−g, −f) = (1, −2)
any point in the xy plane.
3
Activity :
Check whether the following equations Radius of circle r = (−g )2 + (− f )2 − c
represent a circle. If, so then find its centre and
radius. = 16 = 4
9 3
a) x2 + y2 – 6x – 4y + 9 = 0
b) x2 + y2 − 8x + 6y + 29 = 0 Ex. 2) Find the equation of the circle passing
c) x2 + y2 + 7x – 5y + 15 = 0 through the points (5,−6), (1,2) and
(3, −4).
Solution : Conside P = (5, −6), Q = (1, 2),
130
R = (3, −4) C (h, k) Ex. 3) Show that the points (5, 5), (6, 4), (−2, 4)
Fig. 6.6 and (7, 1) are on the same circle; i.e. these
Let the centre of the Q points are concyclic.
circle be at C(h, k) R
∴ r = CP = CQ = CR P Solution :
= radii of the same circle
Let, the equation of the circle be
Consider CP = CQ
x2 + y2 + 2gx + 2fy + c = 0 ....... (I)
∴ CP2 = CQ2
The circle passing through (5, 5), (6, 4),
By using distance formula, (−2, 4)
(h − 5)2 + (k + 6)2 = (h − 1)2 + (k − 2)2 ∴ 50 + 10g + 10f + c = 0 ....... (II)
∴ 52 + 12g + 8f + c = 0 ....... (III)
∴ h2 − 10h + 25 + k2 + 12k + 36 ∴ 20 − 4g + 8f + c = 0 ....... (IV)
= h2 − 2h + 1 + k2 − 4k + 4
i.e −8h + 16k = −56 Now (III) − (IV) gives 32 + 16g = 0
h – 2k = 7 ........ (I) ∴ 16g = −32 ∴ g = –2
Now consider, CQ = CR ∴ CQ2 = CR2 Subtracting (II) from (III)
(h − 1)2 + (k − 2)2 = (h − 3)2 + (k + 4)2 we get 2 + 2g − 2f = 0 ..... (V)
∴ h2 − 2h + 1 + k2 − 4k + 4 substitute g = –2 in equation (V)
= h2 − 6h + 9 + k2 + 8k + 16 2 + (–4) – 2f = 0 2 - 4 = 2f ∴ f = –1
i.e. 4h − 12k = 20 ∴ h − 3k = 5 ......... (II) Now substitute g = −2 and f = −1 in eqn. (IV)
Now, subtracting (II) from I, we get, 20 + c = 4(–2) – 8(1) = −8 + 8
20 + c = 0 ∴ c = –20
k=2 Now substituting g = −2, f = –1 and c = –20 in
Substituting k = 2 in (II) we get
h = 11 ∴ C = (11, 2) equation (I) we get,
x2 + y2 − 4x −2y – 20 = 0 .... (VI) is the
Radius of the circle is equation of the circle passing through the
r = CP = (11− 5)2 + (2 + 6)2 = 100 = 10; points (5,5), (6,4) and (−2,4).
By using centre-radius form, equation of If (7,1) satisfies equation (VI), the four points
the circle is (x − 11)2 + (y − 2)2 = 100 are concyclic.
L.H.S. = (7)2 + (1)2 − 28 − 2 − 20
∴ x2 − 22x + 121 + y2 − 4y + 4 = 100 = 49 + 1 − 50 = 0 = R.H.S.
∴ Thus, the point (7, 1) satisfies the equation of
∴ x2 + y2 − 22x – 4y + 25 = 0 circle.
∴ The given points are concyclic .
131
EXERCISE 6.2 x = h + r cos θ and y = k + r sin θ.
Hence co-ordinates of any point on the circle
(1) Find the centre and radius of each of the are (h + r cos θ , k + r sin θ).
following. (2) Sometimes parametric form is more convenient
(i) x2 + y2 − 2x + 4y − 4 = 0
(ii) x2 + y2 − 6x − 8y − 24 = 0 for calculation as it contains only one variable.
(iii) 4x2 + 4y2 − 24x − 8y − 24 = 0
6.3.1 Tangent : When a line intersects a circle in
(2) Show that the equation coincident points, then that line is called as a
3x2 + 3y2 + 12x + 18y − 11 = 0 represents a tangent of the circle and the point of intersection
circle. is called point of contact.
(3) Find the equation of the circle passing through The equation of tangent to a standard circle
the points (5, 7), (6, 6) and (2, −2). x2 + y2 = r2 at point P(x1, y1) on it.
(4) Show that the points (3, −2), (1, 0), (−1, −2) and Given equation of a circle is x2 + y2 = r2. The
(1, −4) are concyclic.
centre of the circle is at origin O(0, 0) and radius is r.
Let P (x1, y1) be any point on the circle.
6.3 Parametric Form of a circle : Slope of OP = y1 − 0 = y1 , if x1 ≠ 0, y1 ≠ 0.
x1 − 0 x1
Let P(x, y) be any A tangent is drawn to the circle at point P.
point on a circle Since OP is perpendicular
with centre at O to tangent at point P.
slope of the tangent
and radius r. θ P(x1, y1)
As shown in
− x1
Fig. 6.7, OP makes m= y1 O(0, 0)
an angle θ with the ∴ equation of the tangent Fig. 6.8
in slope point form is
positive direction Fig. 6.7
of X-axis. Draw
y – y1 = m(x – x1)
PM ⊥ X-axis from P.
− x1
∆ OMP is a right angled triangle, y – y1 = y1 (x – x1)
OM x PM y yy1 – y12 = –xx1 + x12
∴ cos θ = OP = r ; sin θ = OP = r
∴ x = r cos θ ∴ y = r sin θ xx1 + yy1 = x12 + y12 .............(I)
x = r cos θ and y = r sin θ is the parametric form As (x1, y1) lies on the circle, x12 + y12 = r2,
therefore equation (I) becomes xx1 + yy1 = r2
of circle x2 + y2 = r2. θ is called parameter.
Note that: Thus, In general, equation of the tangent to a
(1) The parametric form of circle
(x − h)2 + (y − k)2 = r2 is given by circle x2 + y2 = r2 at point P(x1, y1) is xx1+yy1= r2
132
Follow the method given above and verify that ∴ −a2m 2 + −a2 2 = a2
equation of tangent to the circle x2 + y2 + 2gx + 2fy c c
+ c = 0 at (x1, y1) is xx1 + yy1 + g(x+x1) + f(y+y1) +
c=0
a4m2 a4 = a2
To find equation of tangent to the curve at c2 + c2
(x1, y1) replace x2 by x x1, 2x by (x + x1), y2 by y
y1, 2y by (y + y1) a2m2 + a2 = c2. i.e. c2 = a2m2 + a2,
Equation of tangent in parametric form. which is the required condition of tangency
Substituting rcosθ1 for x1 and rsinθ1, foy y1, the and the point of contact P(x1, y1) ≡ ( –a2m , a2 ).
equation of a tangent to the circle x2 + y2 = r2 at c c
point P (r cos θ1, r sin θ1) = (x1,y1) is Thus a line y = mx + c is a tangent to the circle
x.r cos θ1 + y.r sin θ1 = r2 i.e. x cos θ1 + y sin θ1 = r
x2 + y2 = a2, if c2 = a2m2 + a2 i.e. c = ± a2m2 + a2
and the point of contact is –a2m , a2 .
c c
Thus, there are two tangents with the same
6.3.2 Condition of tangency : slopem,y= mx + a2 (1+ m2 ) , y = mx − a2 (1+ m2 )
To find the condition that a line y = mx + c is To check the tangency of a straight line to a
a tangent to the circle x2 + y2 = a2 and also to find circle, it is enough to show that the perpendicular
the point of contact, from the center to the line is equal to the radius.
Let the equation of the line be y = mx + c 6.3.3 Tangents from a point to the circle
\ mx – y + c = 0 ..........(I)
Equation of a tangent to the circle x2 + y2 = a2
at point (x1 y1) on it is xx1 + yy1 = a2
i.e. x1 x + y1 y – a2 = 0 .............(II)
If the line given by equation (I) is tangent
to the circle then equation (I) and equation (II)
represent the same (tangent) line.
Comparing the co-efficients of the like terms Fig. 6.9
in these equations, From any point outside the circle and in the
same plane, two tangents can be drawn to the
x1 = y1 = –a2 circle.
m –1 c
Let P(x1,y1) be a point in the plane, outside
∴ x1 = –a2 and y1 = –a2 the circle.
m c –1 c
If a tangent from P to the circle has slope
∴ x1 = –a2m and y1 = a2 m, the equation of the tangent is y−y1 = m(x−x1)
c c i.e. mx−y1−mx1+y1 = 0.
But the point (x1, y1) lies on the circle
∴ x12 + y12 = a2
133
The condition that this is tangent to the circle
is y1 − mx1 = a, the radius.
1+ m2
∴ (y1−mx1)2 = a2(1+m2)
∴ x2−2x1y1m + x12m2 = a2+a2m2
∴ (x12−a2) m2−2x1y1m + (y12 − a2) = 0 is
quadratic equation in m.
It has two roots say m1 and m2, which are the
slopes of two tangents.
Thus two tangents can be drawn to a circle Fig. 6.10
from a given point in its plane.
Sum of the roots (m1 + m2) = –(–2x1 y1) SOLVED EXAMPLES
(x12 – a2)
= 2x1 y1 Ex. 1) Find the parametric equation of the circle
x12 – a2 x2 + y2 − 6x + 4y − 3 = 0
Solution : We write the equation of the circle as
Product of the roots (m1 m2) = (y12 – a2) (x2 − 6x + 9) + (y2 + 4y + 4) = 3 + 9 + 4
(x12 – a2) (x − 3)2 + (y + 2)2 = 16
(x − 3)2 + (y + 2)2 = (4)2
6.3.4 Director Circle: The parmetric equations are
x − 3 = 4 cos θ and y + 2 = 4 sin θ
The locus of the point of intersection of that is, x = 3 + 4 cos θ and y = −2 + 4 sinθ
perpendicular tangents to a circle
If the tangents drawn from a point P are
mutually perpendicular to each other them
m1 m2 = – 1 and we have, Ex. 2) Show that the line 3x − 4y + 15 = 0 is a tangent
to the circle x2 + y2 = 9. Find the point of
y12 – a2 =–1 contact.
x12 – a2
∴ y12 – a2 = – (x12 – a2) Solution : The equation of circle is x2 + y2 = 9
∴ y12 – a2 = – x12 – a2
∴ x12 + y12 = a2 + a2 ...(1)
∴ x12 + y12 = 2a2.
The equation of line is 3x − 4y + 15 = 0 ...(2)
Which represents a circle and is called as
equation of the director circle of the circle ∴y= 1 (3x + 15), substitute value of y in
4
x2 + y2 = a2.
equation (1)
x2 + 1 (3x + 15)2 = 9
16
134
∴ 16x2 + 9x2 + 90x + 225 = 144 Ex. 3) Find the equation of the tangent to the circle
x2 + y2 − 4x − 6y − 12 = 0 at (−1, −1)
∴ 25x2 + 90x + 81 = 0
Solution : The equation of circle is
∴ (5x + 9)2 = 0 x2 + y2 − 4x − 6y − 12 = 0
It is of the type x2 + y2 + 2gx + 2fy + c = 0
−9
\ g = −2, f = −3, c = −12
x = 5 ; The roots of equation are equal. Let P (−1, −1) = (x1, y1)
\ line (2) is tangent to given circle (1) We know that the equation of a tangent to
a circle
\ y = 1 (3x + 15) x2 + y2 + 2gx + 2fy + c = 0 at (x1, y1) is
xx1 + yy1 + g(x + x1) + f(y+y1) + c = 0
4 x(−1) + y(−1) + 2(x−1) −3 (y−1) −12 = 0
= 1 (3 − 9 + 12) −x −y −2x + 2 −3y + 3−12 = 0
4 5
3x + 4y + 7 = 0
= 12
5
− 9 , 12 is the only point of intersection of
3 5
the line and circle.
\ The line 3x − 4y +15 = 0 touches the circle
at − 9 ,12
35
\ point of contact = − 9 , 12
3 5
Activity : EXERCISE 6.3
Equation of a circle is x2 + y2 = 9. (1) Write the parametric equations of the circles
Its centre is at ( , ) and radius is (i) x2 + y2 = 9 (ii) x2 + y2 + 2x − 4y − 4 = 0
Equation of a line is 3x – 4y + 15 = 0 (iii) (x − 3)2 + (y + 4)2 = 25
(2) Find the parametric representation of the
∴y= x+ circle 3x2 + 3y2 − 4x + 6y − 4 = 0.
(3) Find the equation of a tangent to the circle
Comparing it with y = mx + c x2 + y2 − 3x + 2y = 0 at the origin.
(4) Show that the line 7x − 3y − 1 = 0 touches the
m = and c = circle x2 + y2 + 5x − 7y + 4 = 0 at point (1, 2)
(5) Find the equation of tangent to the circle
We know that, if the line y = mx + c is a tangent x2 + y2 − 4x + 3y + 2 = 0 at the point (4, −2)
to x2 + y2 = a2 then c2 = a2m2 + a2
Hence c2 = ( )2 Also, c2 = a2m2 + a2
225
= 16 .... (I) =9 +9
=9 9 + 1
16
= 9(25)
16
= 225 ...........(II)
16
From equations (I) and (II) we conclude that
135
Let's Remember MISCELLANEOUS EXERCISE - 6
1) Equation of a standard circle is x2 + y2 = r2. (I) Choose the correct alternative.
Its centre is at (0, 0) and radius is r.
(1) Equation of a circle which passes through
2) Equation of a circle in (centre-radius) form (3, 6) and touches the axes is
is (A) x2 + y2 + 6x + 6y + 3 = 0
(B) x2 + y2 − 6x − 6y − 9 = 0
(x – h)2 + (y – k)2 = r2 (C) x2 + y2 − 6x − 6y + 9 = 0
Its centre is at (h, k) and radius is r. (D) x2 + y2 − 6x + 6y − 3 = 0
3) Equation of a circle in general form is (2) If the lines 2x − 3y = 5 and 3x − 4y = 7 are the
diameters of a circle of area 154 sq. units,
x2 + y2 + 2gx + 2fy + c = 0. then find the equation of the circle.
(A) x2 + y2 − 2x + 2y = 40
Its centre is at (–g, –f) and radius is (B) x2 + y2 − 2x − 2y = 47
g2 + f 2 – c (C) x2 + y2 − 2x + 2y = 47
(D) x2 + y2 − 2x − 2y = 40
If g2 + f 2 > c then the circle is real, it can be
drawn in the xy plane. (3) Find the equation of the circle which passes
through the points (2, 3) and (4, 5) and the
If g2 + f 2 = c then the circle reduces to a centre lies on the straight line y − 4x + 3 = 0.
point. (A) x2 + y2 − 4x − 10y + 25 = 0
(B) x2 + y2 − 4x − 10y − 25 = 0
If g2 + f 2 < c then the circle is not real and it (C) x2 + y2 − 4x + 10y − 25 = 0
cannot be drawn in xy plane. (D) x2 + y2 + 4x − 10y + 25 = 0
4) Equation of a standard circle x2 + y2 = r2 in
parametric form is x = rcosq y = rsinq.
5) Equation of a tangent to the circle
x2 + y2 = r2 at point (x1 y1) on it is xx1 + yy1 = r2
in the Cartesian form and point of contact is
–r2m , r2 . Equation of a tangent in (4) The equation of the tangent to the circle
c c x2 + y2 = 4 which are parallel to
x + 2y + 3 = 0 are
parametric form to the circle x2 + y2 = r2
at point P(x1, y1) ≡ P(q1), where q1 is the
parameter, is cosq1 x + sinq1 y = r is. (A) x − 2y = 2 (B) x + 2y = ± 2 3
6) A line y = mx + c is a tangent to the circle (C) x + 2y = ± 2 5 (D) x − 2y = ± 2 5
x2 + y2 = r2 if and only if c2 = r2m2 + r2.
(5) If the lines 3x − 4y + 4 = 0 and
7) Equation of a tangent line in terms of slope 6x − 8y − 7 = 0 are tangents to a circle, then
is y = mx ± r2m2 + r2 . find the radius of the circle
8) Equation of director circle of circle (A) 3 (B) 4 (C) 1 (D) 7
x2 + y2 = r2 is x2 + y2 = 2r2. 4 3 4 4
136
(6) Area of the circle centre at (1, 2) and passing (II) Answer the following :
through (4, 6) is
Q. 1 Find the centre and radius of the circle
(A) 5π (B) 10π x2 + y2 − x +2y − 3 = 0
C) 25π (D) 100π
Q. 2 Find the centre and radius of the circle
(7) If a circle passes through the point (0, 0), x = 3 – 4 sinq, y = 2 – 4cosq
(a, 0) and (0, b) then find the co-ordinates of
its centre. Q. 3 Find the equation of circle passing through
the point of intersection of the lines
(A) −a , −b (B) a , −b x + 3y = 0 and 2x − 7y = 0 whose centre
2 2 2 2 is the point of intersection of lines
x + y + 1 = 0 and x − 2y + 4 = 0.
a b
(C) −a , b (D) 2 , 2 Q. 4 Find the equation of circle which passes
2 2 through the origin and cuts of chords of
length 4 and 6 on the positive side of
(8) The equation of a circle with origin as centre x - axis and y axis respectively.
and passing through the vertices of an
equilateral triangle whose median is of Q. 5 Show that the points (9, 1), (7, 9), (−2, 12)
length 3a is and (6, 10) are concyclic.
(A) x2 + y2 = 9a2 (B) x2 + y2 = 16a2 Q. 6 The line 2x − y + 6 = 0 meets the circle
(C) x2 + y2 = 4a2 (D) x2 + y2 = a2 x2 + y2 + 10x + 9 = 0 at A and B. Find the
equation of circle on AB as diameter.
(9) A pair of tangents are drawn to a unit circle
with cnetre at the origin and these tangents Q. 7 Show that x = −1 is a tangent to circle
intersect at A enclosing an angle of 60 . The x2 + y2 − 2y = 0 at (−1, 1).
area enclosed by these tangetns and the area
of the circle is Q. 8 Find the equation of tangent to the circle
(A) 2 − π (B) 3 − π x2 + y2 = 64 at the point P 2π
3 6 3 3
(C) π − 3 (D) 3 1 − π Q.9 Find the equation of locus of the point of
3 6 6 intersection of perpendicular tangents
drawn to the circle x = 5cosq and y = 5sinq.
(10) The parametric equations of the circle
x2 + y2 + mx + my = 0 are Q.10 Find the equation of the circleconcentricwith
x2 + y2 – 4x + 6y = 1 and having radius
(A) x = −m + m cosθ , y= −m + m sinθ 4 units.
2 2 2 2
(B) x = −m + m cosθ , y= +m + m sinθ Q.11 Find the lengths of the intercepts made on
2 2 2 2 the co-ordinate axes, by the circle.
(C) x = 0 , y = 0 (i) x2 + y2 – 8x + y – 20 = 0
(ii) x2 + y2 – 5x + 13y – 14 = 0
(D) x = mcosθ ; y = msinθ
137
Q.12 Show that the circles touch each other Q.21 Find the equations of the tangents to the
externally. Find their point of contact and circle x2 + y2 = 16 with slope –2.
the equation of their common tangent.
Q.22 Find the equations of the tangents to the
i) x2 + y2 – 4x + 10y +20 = 0, circle x2 + y2 = 4 which are parallel to
x2 + y2 + 8x – 6y – 24 = 0. 3x + 2y + 1 = 0.
ii) x2 + y2 – 4x – 10y + 19 = 0,
x2 + y2 + 2x + 8y – 23 = 0. Q.23 Find the equations of the tangents to the
circle x2 + y2 = 36 which are perpendicular
Q.13 Show that the circles touch each other to the line 5x + y = 2.
internally. Find their point of contact and
the equation of their common tangent. Q.24 Find the equations of the tangents to the
circle x2 + y2 – 2x + 8y – 23 = 0 having
i) x2 + y2 – 4x – 4y – 28 = 0, slope 3.
x2 + y2 – 4x – 12 = 0.
ii) x2 + y2 + 4x – 12y + 4 = 0, Q.25 Find the eqn of the locus of a point, the
x2 + y2 – 2x – 4y + 4 = 0. tangents from which to the circle x2 + y2 = 9
are at right angles.
Q.14 Find the length of the tangent segment
drawn from the point (5, 3) to the circle Q.26 Tangents to the circle x2 + y2 = a2 with
x2 + y2 + 10x – 6y – 17 = 0. inclinations, q1 and q2 intersect in P. Find
the locus of such that
Q.15 Find the value of k, if the length of the
tangent segment from the point (8, –3) to i) tan q1 + tan q2 = 0 ii) cot q1 + cot q2 = 5
the circle. iii) cot q1. cot q2 = c.
x2 + y2 – 2x + ky – 23 = 0 is 10 . Extra Information :
Q.16 Find the equation of tangent to Circle 1)
x2 + y2 – 6x – 4y = 0, at the point (6, 4) on it.
Q.17 Find the equation of tangent to Circle
x2 + y2 = 5, at the point (1, –2) on it.
Q.18 Find the equation of tangent to Circle
x = 5 cosq. y = 5 sinq, at the point q = π/3
on it.
Q.19 Show that 2x + y + 6 = 0 is a tangent to Fig. 6.10
x2 + y2 + 2x – 2y – 3 = 0. Find its point of
contact. Circles touching each other externally.
d(c1 c2) = r1 + r2.
Q.20 If the tangent at (3, –4) to the circle Exactly three common tangents can be drawn.
x2 + y2 = 25 touches the circle x2 + y2 + 8x –
4y + c = 0, find c.
138
2) 4)
Fig. 6.11 Fig. 6.13
Circles touching each other internally. Circles intersecting each other.
d(c1 c2) = | r1 − r2 | Line joining the point of intersection is the
Exactly one common tangent can be drawn. common chord also called as the radical axis.
(seg AB)
3) Exactly two common tangent can be drawn.
5)
Fig. 6.12 Fig. 6.14
Disjoint circles. Concentric circles
| r1 + r2 | < d(c1 c2) No common tangent can be drawn.
Exactly four common tangents can be drawn.
139
7 Conic sections
Let's Learn a wide range of applications such as planetary
motions, in designs of telescopes and antennas,
Conic sections : parabola, ellipse, hyperbola reflection in flash light, automobile headlights,
Standard equation of conics construction of bridges, navigation, projectiles
Equation of tangent to the conics etc.
Condition for tangency
A straight line, a circle, parabola, ellipse and
Let's Recall hyperbola are all conic sections. Of these we have
studied circle and straight line.
Section formulae : Let A (x1,y1) and
Earlier we have studied different forms of
B(x2,y2) be two points in a plane. If P equations of line, circle, and their properties. In
and Q divide seg AB in the ratio m:n this chapter we shall study some more curves,
internally and externally respectively then namely parabola, ellipse and hyperbola which are
conic sections.
P= mx2 + nx1 , my2 + ny1 and Let's Learn
m + n m+n
7.1.1 Double cone:
Q = mx2 − nx1 , my2 − ny1
m−n m−n Let l be a fixed
line and m another line
Introduction: intersecting it at a fixed
point V and inclined
The Greek mathematician Archimedes at an acute angle θ Fig.7.1
and Apollonius studied the curves named conic (fig 7.1.) Suppose we
sections. These curves are intersections of a plane rotate the line m around
with right circular cone. Conic sections have the line l in such a way
that the angle θ remains
V
Fig.7.2 Fig.7.3(a) Fig.7.3(b) Fig.7.4
140
constant. Then the surface generated is a double- This constant ratio is Fig. 7.5
napped right circular cone. called the eccentricity of the
conic section, denoted by e.
The point V is called the vertex, the line l
is called axis, and the rotating line m is called a Hence we write SP = e
generator of the cone. The vertex V separates the PM
cone in to two parts called nappes (Fig.7.2).
or SP = e PM. This is called
7.1.2. Conic sections : Focus - Directrix property
of the conic section.
Let’s construct
The nature of the conic section
Take a carrot or a cone made of drawing paper depends upon the value of e.
and cut it with a plane satisfying the following
conditions. i) If e = 1, the conic section is called parabola.
i) The plane is perpendicular to the axis and ii) If 0 < e < 1, the conic section is called an
does not contain vertex, the intersection is a ellipse.
circle (studied earlier Fig. 7.3(a)).
iii) If e > 1, the conic section is called
ii) The plane is parallel to one position of the hyperbola.
generator but does not pass through the
vertex, we get a parabola (Fig. 7.3(a)). 7.1.4. Some useful terms of conic sections:
iii) The plane is oblique to the axis and not 1) Axis: A line about which a conic section
parallel to the generator we get an ellipse is symmetric is called an axis of the conic
(Fig. 7.3(a)). section.
iv) If a double cone is cut by a plane parallel to 2) Vertex : The point of intersection of a conic
axis, we get parts of the curve at two ends section with its axis of symmetry is called a
called hyperbola (Fig. 7.3(b)). vertex.
v) A plane containing a generator and tangent 3) Focal Distance : The distance of a point on
to the cone, intersects the cone in that a conic section from the focus is called the
generator. We get pair to straight lines focal distance of the point.
(Fig. 7.4).
7.1.3. Definition of a conic section and its 4) Focal chord : A chord of a conic section
equation: 5) passing through its focus is called a focal
chord.
A conic section or conic can be defined as
the locus of the point P in a plane such that the Latus-Rectum: A focal chord of a conic
ratio of the distance of P from a fixed point to its section which is perpendicular to the axis of
distance from a fixed line is constant. symmetry is called the latus-rectum.
The fixed point is called the focus of the conic 6) Centre of a conic : The point which bisects
section, denoted by S. The fixed straight line is every chord of the conic passing through it,
called the directrix of conic section, denoted by d. is called the centre of the conic.
If S is the focus, P is any point on the 7) Double ordinate : A chord passing through
conic section and segment PM is the length of any point on the conic and perpendicular to
perpendicular from P on the directrix, then by the axis is called double ordinate.
definition SP = constant. (fig. 7.5)
PM
141
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