Coordinate Geometry

Unit 8

COORDIANTE GEOMETRY

REVIEW Tara Bdr Magar
x2  x1 2  y2  y1 2
1. Distance formula:

The distance between two points A (x1, y1) and B (x2, y2) is given by d =

2. Mid-point formula:

The mid-point of line segment joining the points A (x1, y1) and B (x2, y2) is given by

(x, y) =  x1  x2 , y1  y2 
 2 2 

3. Section formula for internal division:

The coordinates of a point which divides the line segment joining the points A (x1, y1) and B (x2, y2)

internally in the ration m1:m2 is given by (x, y) =  m1x 2  m2x1 , m1 y 2  m2 y1 
m1  m2 m1  m2

4. Section formula for external division:
The coordinates of a point which divides the line segment joining the points A (x1, y1) and B (x2, y2)

externally in the ration m1:m2 is given by (x, y) =  m1x 2  m2x1 , m1 y 2  m2 y1 
m1  m2 m1  m2

5. Centroid formula:

The coordinates of centroid of triangle ABC with vertices A (x1, y1), B (x2, y2) and C (x3, y3) is given by

G (x, y) =  x1  x2  x3 , y1  y2  y3 
 3 3 

6. Slope formula:

(i) The slope of the line segment joining the points A (x1, y1) and B (x2, y2) is given by m = y2  y1
x2  x1

(ii) If the line makes an angle of  with x-axis then slope (m) = tan  [useful in class 10]

(iii) The slope of the line ax + by + c = 0 is given by m =  Coefficine t of x   a [useful in cl-10]
Coefficine t of y b

7. Equation of line:
(i) Equation of x-axis is given by y = 0
(ii) Equation of y-axis is given by x = 0
(iii) Equation of line in slope intercept form is given by y = mx + c

(iv) Equation of line in double intercept form is given by x  y 1
a b

(v) Equation of line in perpendicular form is given by xcosα  ysinα  p

(vi) Equation of line in slope-point form is given by y – y1 = m (x – x1) [useful in class 10]

(vii) Equation of line in two-point form is given by y – y1 = y2  y1 (x – x1)
x2  x1

Course for Class 10 K- Level U-Level A-Level HA-Level Total Marks
21=2
SEE SPECIFICATION GRID 22= 4 14=1 15=5 2+4 + 4 + 5 = 15
Contents
4.1 Angle between two lines
4.2 Pair of straight lines
4.3 Conic section
4.4 Circle

4.1 Angle between two straight lines

1. Find the angle between two straight lines y = m1x + c1 and y = m2x + c2
Solution:
Let AB and CD be two straight lines with equations y
= m1x + c1 and y = m2x + c2 respectively.
Let the lines AB and CD cut the x-axis at E and F

respectively by making angles 1 and 2 in positive

direction. Then, slope of line AB (m1) = tan1 and

slope of line CD (m2) = tan2

Let AB and CD intersect at G so that BGD = and

BGC = 1800 – .
Now,

By plane geometry,  + 2 = 1

  = 1 – 2
Taking tan on both sides, we get

tan  = tan (1 – 2 )

 tanθ1  tanθ2
1 tanθ1.tanθ2

 m1  m2
1  m1.m2

BGD () = t an1 m1  m2  … (i)
1 m1.m2

BGC = 1800 –  is also the angle between AB and CD.

So, tan BGC = tan (1800 –) = – tan   m1  m2
1 m1.m2

BGC = tan1  m1  m2  … (ii)
1  m1.m2

Hence, from (i) and (ii); the angle between the lines is t an 1  m1  m2  .
1 m1.m2

1. If the angle between two straight lines with slopes m1 and m2 is , then t anθ   m1  m2
1 m1.m2

2. If the value of tan is positive then  will be acute.

3. If the value of tan is negative then  will be obtuse.
4. If the value of tan is 0 then m1 = m2 and the lines are parallel.
5. If the value of tan is undefined then m1. m2 = -1 and the lines are perpendicular.

Homework: Find the angle between two straight lines y = m1x + c1 and y = m2x + c2

Condition of lines being parallelism

If the lines y = m1x + c1 and y = m2x + c2 are parallel to each other, then angle between them () = 00.

Now,

t anθ   m1  m2
1 m1m2

or, tan00   m1  m2
1  m1m2

or, 0   m1  m2
1 m1m2

or, m1  m2  0

m1  m2

Thus, the lines are parallel (coincident) if and only if (iff) their slopes are equal i.e., m1 = m2

Condition of lines being perpendicularity

If the lines y = m1x + c1 and y = m2x + c2 are perpendicular to each other, then angle between them () = 900.

Now,

t anθ   m1  m2
1 m1m2

or, tan900   m1  m2
1  m1m2

or, sin900   m1  m2
cos900 1 m1m2

or, 1   m1  m2
0 1  m1m2

or,1  m1m2  0

m1.m2  1

Thus, the lines are perpendicular (orthogonal) to each other iff the product of their slopes is

equal to – 1. i.e., m1×m2 = - 1.

KNOWLEDGE (K) BASED PROBLEMS

1. (a) If the slopes of two straight lines are m1 and m2 respectively and  be the angle between

them, write the formula for tan. [SEE MODEL-2076]

(b) Write the formula to find the angle between the lines y = m1x + c1 and y = m2x + c2.

(c) Write the formula to find the angle between the lines y = m1x and y = m 2x.

2. (a) If two straight lines y = m1x + c1 and y = m2x + c2 are parallel to each other, write the
relation between m1 and m2.

(b) If two straight lines y = m1x + c1 and y = m2x + c2are perpendicular (orthogonal) to each
other, write the relation between m1 and m2.

3. (a) In the straight lines l1: y = m1x + c1 and l2: y = m2x + c2, if m1 = m2, write the relation
between the lines l1 and l2.

(b) In the straight lines l1: y = m1x + c1 and l2: y = m2x + c2, if m1. m2= -1, what kind of lines
l1 and l2?

4. (a) Find the slope of the straight line passing through the points (x1, y1) and (x2, y2).
(b) Find the slope of the straight line passing through the points (2, - 1) and (-6, 3).

5. (a) Find the slope of the line having equation ax + by + c = 0.
(b) Find the slope of the line having equation 2x + 3y = 5.
(c) Find the slope of the line having equation y = 5.
(d) Find the slope of the line having equation x = 1.

6. (a) If the slope of a side AB of a rectangle ABCD is 1/2,what is the slope of its opposite side
CD?

(b) If the slope of a side of a square is 3/4, what is the slope of its opposite side?

7. (a) If the slope of a diagonal AC of a square ABCD is 3 what is the slope of its diagonal BD?
(b) If the slope of a diagonal of a rhombus is 3 , what is the slope of its another diagonal?