Chapter 4: CONICS

Lecture Note: 1 of 4 Mathematics 2 SM025
Ch. 4 Conics Session 2021/2022

Topic: 4. CONICS

Sub-Topic: 4.3 Parabolas

Learning Outcomes: At the end of this lesson, students should be able to
(a) find the equation of the parabola.
(b) determine the vertex, focus and directrix of a parabola by completing the square.

The Equation of a Parabola
A graph of parabola looks like a big . Here are the important names:

Opens upward and downward Opens left and right

Vertex, h,k Vertex, h,k
Focus, h,k  p Focus, h  p,k

Directrix, y  k  p Directrix, x  h  p

Equation, x  h2  4p y  k Equation, y  k2  4p x  h

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Lecture Note: 1 of 4 Mathematics 2 SM025
Ch. 4 Conics Session 2021/2022

EXAMPLE 1
State the vertex, focus and directrix for each of the following:

(a) y  22  12x  3. (b) x  12  5y  2.

(a) y  22  12x  3

y  k2  4p x  h

4p  12
p30

V h,k  V 3,2

F h  p,k  F 6,2

Directrix; x  h  p
 33

x0

(b) x  12  5y  2

x  h2  4p y  k

4p  5
p 50
4

V h,k  V 1,2

F h,k  p  F 1, 34

Directrix; y kp

 2  5
4

y   13
4

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Lecture Note: 1 of 4 Mathematics 2 SM025
Ch. 4 Conics Session 2021/2022

EXAMPLE 2
Write down the equation of given parabola below in standard form. For each parabola, state the
coordinates of the vertex, focus and the equation of the directrix. Hence, sketch each graph.

(a) x2  8x  4y  12  0. (b) y2  8y  2x  22  0.

(a) x2  8x  4y  12

x2  8x  4y  12

x  42  42  4y  12
x  42  4y  4
x  42  4y  1
x  h2  4p y  k

4p  4 0
p  1  0

V h,k  V 4,1

F h,k  p  F 4,0

Directrix; y  k  p

 1  1

y2

(b) y2  8y  2x  22 y

y  42  2x  22  16 0 x
y  42  2x  6
y  42  2x  3 V 3,4 F 72 ,4
y  k2  4px  h

4p  2
p 1 0
2

V h,k  V 3,4

F h  p,k  F 72 ,4

Directrix; xhp

31
2

x 5
2

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Lecture Note: 1 of 4 Mathematics 2 SM025
Ch. 4 Conics Session 2021/2022

EXAMPLE 3

Find the equation of a parabola, which satisfies the following conditions, vertex 1,2, its axis

parallel to the y-axis and the parabola passes through the point 3,6.

x  h2  4p y  k
x  12  4p y  2

At point 3,6;

3  12  4p 6  2

16  32p
p1

2

x  12  412y  2
x  12  2y  2

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Lecture Note: 2 of 4 Mathematics 2 SM025
Ch. 4 Conics Session 2021/2022

Topic: 4. CONICS

Sub-Topic: 4.2 Ellipses

Learning Outcomes: At the end of this lesson, students should be able to
(a) determine the equation of the ellipse.
(b) determine all the vertices, foci, major and minor axes.
(c) determine the centre and foci of an ellipse by completing the square.

The Equation of an Ellipse

For b > a

c2  b2  a2

Centre, C  h,k

Equation of the major axis, x  h
Equation of the minor axis, y  k

Foci, F  h, k  c

Vertices on major axis, V  h,k  b

Vertices on minor axis, V  h  a,k

Equation of ellipse, x  h2  y  k2 1

a2 b2

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Lecture Note: 2 of 4 Mathematics 2 SM025
Ch. 4 Conics Session 2021/2022

For a > b

c2  a2  b2

Centre, C  h,k

Equation of the major axis, y  k
Equation of the minor axis, x  h

Foci, F  h  c, k

Vertices on major axis, V  h  a,k

Vertices on minor axis, V  h,k  b

Equation of ellipse, x  h2 y  k2 1

a2  b2

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Lecture Note: 2 of 4 Mathematics 2 SM025
Ch. 4 Conics Session 2021/2022

EXAMPLE 4

Find the centre and the foci of the ellipse x  32  y  12  1. Hence, sketch the graph.

94

x  h2 y  k2

a2  b2  1
a2  9, a  3, b2  4;

c  a2  b2
 94
5

C h,k  C 3,1

F h  c,k  F 3  5,1, F 3  5,1

V h  a,k  V 6,1, V 0,1

V 6,1 V 0,1
0

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Lecture Note: 2 of 4 Mathematics 2 SM025
Ch. 4 Conics Session 2021/2022

EXAMPLE 5
Sketch the ellipse with equation 25x2  16y2  50x  375  0

25x2  16y2  50x  375  0

25x2  2x  16y2  375

 25 x  12  1  16y2  375

25x  12  16y2  400
x  12  y2  1

16 25
a2  16, b2  25, b  5;

c  b2  a2
 25  16
3

C h,k  C 1,0

F h,k  c  F 1,3, F 1,3

V h,k  b  F 1,5, F 1,5

0

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Lecture Note: 2 of 4 Mathematics 2 SM025
Ch. 4 Conics Session 2021/2022

EXAMPLE 6

Find the equation of an ellipse with centre 3,1 and the major axis running parallel with the

y-axis. The length of the major axis is 10 units and the minor axis is 6 units. Sketch the ellipse.

x  32  y  12  1

9 25
b2  25, a2  9;

c  b2  a2
 25  9
4

C h,k  C 3,1

F h,k  c  F 3,3, F 3,5

V h,k  b  V 3,4, V 3,6

0

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Lecture Note: 2 of 4 Mathematics 2 SM025
Ch. 4 Conics Session 2021/2022

EXAMPLE 7

The center of an ellipse is 1,2, the minor axis is parallel to the y-axis and the ellipse passes

through points 4,2 and 5, 52.Find the general equation of the ellipse. Sketch the ellipse.

x  h2 y  k2

a2  b2  1

C 1,2,

x  12  y  22 1
b2
a2

At 4,2,

4  12  2  22 1
b2
a2

52 1
a2

a2  25

At 5, 52,

5  12   2 22 1
5
25 b2

42   12 2 1
5
25
b2

b2  16

x  12  y  22  1

25 16

16x  12  25y  22  400

16x2  25y2  32x  100y  284  0

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Lecture Note: 2 of 4 Mathematics 2 SM025
Ch. 4 Conics Session 2021/2022

0

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Lecture Note: 3 of 4 Mathematics 2 SM025
Ch. 4 Conics Session 2021/2022

Topic: 4. CONICS

Sub-Topic: 4.1 Circles

Learning Outcomes: At the end of this lesson, students should be able to
(a) determine the equation of a circle.
(b) determine the centre and radius of a circle.

Introduction

Circle
o A circle is a set of all points in a plane equidistant from a given fixed point, called the

centre, C.

Radius
o Letter r denotes radius of the circle.

Diameter
o Segment AB is the diameter of the circle. It is the maximum

distance between two opposite points on the circle.

Chord
o The chord of a circle is the one that divides the circle into two

parts called major segment and minor segment.

Tangent
o A line that touches the circle at one and only one point is called

as tangent to the circle.
o At any given point of a circle, there is one and only one tangent.
o The tangent at any point of a circle is always perpendicular to

the radius through the point of contact.

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Lecture Note: 3 of 4 Mathematics 2 SM025
Ch. 4 Conics Session 2021/2022

Standard and General Equa tion of a Circle

Standard Equation General Equation

C h,k By expanding x  h2  y  k2  r2,
x  h2  y  k2  r2
x2  2hx  h2  y2  2ky  k2  r2
x2  y2  2hx  2ky  h2  k2  r2  0
Let h  g, k  f, c  h2  k2  r2, thus
x2  y2  2gx  2fy  c  0, where

Centre, C g,f  and Radius, r  g2  f 2  c

Equation of a Circle

Given the radius and the centre. (a) Use the formula x  h2  y  k2  r2.

C h,k

Passing through two points where the (a) Substitute the two points into
equation of the diameter is given. x2  y2  2gx  2fy  c  0.

(b) Substitute centre, g,f  into

ax  by  c  0.
(c) Solve all the three equations simultaneously.

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Lecture Note: 3 of 4 Mathematics 2 SM025
Ch. 4 Conics Session 2021/2022

Passing through three points. (a) Substitute all the three points into
x2  y2  2gx  2fy  c  0.

(b) Solve all the three equations simultaneously.

Given the centre and the tangent. (a) Use the formula x  h2  y  k2  r2.

ax  by  c  0 (b) Find r using formula shortest distance

C h,k r  ah  bk  c , where ax  by  c  0.
a2  b2

EXAMPLE 8

Find the center and radius of the circle x  32  y  52  36.

By comparing x  32  y  52  62 and x  h2  y  k2  r2,
C h,k  C 3,5

r6

EXAMPLE 9
Find the center and radius of the circle 2x2  2y2  20x  8y  14  0.

2x2  2y2  20x  8y  14  0
x2  y2  10x  4y  7  0

By comparing x2  y2  10x  4y  7  0 and x2  y2  2gx  2fy  c  0,
2g  10, 2f  4, c  7

g  5 f  2

C g,f  C 5,2

r  g2  f2  c

 52  22  7

6

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Lecture Note: 3 of 4 Mathematics 2 SM025
Ch. 4 Conics Session 2021/2022

EXAMPLE 10

Find the equation of the circle which passes through the points A 1,3,B 1,1 and has its

diameter on the line x  2y  1.

x2  y2  2gx  2fy  c  0

Since A1,3,B 1,1 lies on the circle

1  9  2g  6f  c  0

2g  6f  c  10.........1

1  1  2g  2f  c  0

 2g  2f  c  2.............2

C g,f  at x  2y  1

 g  2f  1

 f  1, g  3, c  10
 x2  y2  6x  2y  10  0

EXAMPLE 11

Find the equation of the circle having centre 1, 2 and touch the line 2x  y  2  0.

By comparing 2x  y  2  0 and ax  by  c  0,

a  2, b  1, c  2

r  shortest distance from C 1,2 to the line 2x  y  2  0

ah  bk  c
r

a2  b2

21  12  2



22  12

6
5

x  h2  y  k2  r2

x  12  y  22   6 2
5

x  12  y  22  36

5

5x  12  5y  22  36

5x2  5y2  10x  20y  11  0

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Lecture Note: 3 of 4 Mathematics 2 SM025
Ch. 4 Conics Session 2021/2022

EXAMPLE 12

Find the equation of the circle which passes through A 1, 8, B 6,1 and C 2,1.

x2  y2  2gx  2fy  c  0
1  64  2g  16f  c  0

2g  16f  c  65
36  1  12g  2f  c  0

 12g  2f  c  37
4  1  4g  2f  c  0

 4g  2f  c  5
 14g  14f  28
 8g  4f  32

f  2g  8

14g  142g  8  28

14g  28g  112  28
42g  84

g2
 f  4, c  5

 x2  y2  4x  8y  5  0

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Lecture Note: 4 of 4 Mathematics 2 SM025
Ch. 4 Conics Session 2021/2022

Topic: 4. CONICS

Sub-Topic: 4.1 Circles

Learning Outcomes: At the end of this lesson, students should be able to
(c) find the points of intersection of two circles, and a line and a circle.
(d) find the equation of tangents and normal to a circle.
(e) find the length of a tangent from a point to a circle.

The Points of Intersection of Two Circles

Solving the simultaneous equation to find the intersection of two
circles.

The type of intersection between two circles can be determine by using
discriminant b2  4ac OR distance between the two centres

Does NOT intersect. (a) b2  4ac  0
(b) Distance C1C2  r1  r2

Intersect at ONE point. (a) b2  4ac  0
(b) Distance C1C2  r1  r2

(a) b2  4ac  0
Intersect at TWO points.

(b) Distance C1C2  r1  r2

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Lecture Note: 4 of 4 Mathematics 2 SM025
Ch. 4 Conics Session 2021/2022

EXAMPLE 13
Determine the type of intersections between the two circles below:

x  12  y2  1
x  12  y2  1.

x  12  y2  1 When x  0,

x2  2x  1  y2  1 02  y2  20  0
x2  y2  2x  0
y2  0
x  12  y2  1
b2  4ac  02  410
x2  2x  1  y2  1
x2  y2  2x  0 0
Since b2  4ac  0, therefore the two circles intersect
: 4x  0
x0 at one point, i.e. 0,0.

EXAMPLE 14
Find the points of intersection of the two circles with equations x2  y2  3x  13y  48  0 and

x2  y2  x  3y  0.

x2  y2  3x  13y  48  0

x2  y2  x  3y  0
: 4x  16y  48  0

x  4y  12
Substitute  into ,

4y  122  y2  4y  12  3y  0

16y2  96y  144  y2  4y  12  3y  0
17y2  95y  132  0

y  317y  44  0

y  3, y  44
17

When y  3, When y  44 ,
17

x  43  12 x  41474  12
  28
0 17

 the point of intersection are 0, 3 and  28 , 44 .
17 17

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Lecture Note: 4 of 4 Mathematics 2 SM025
Ch. 4 Conics Session 2021/2022

The Points of Intersection of a Line and a Circle

The line and the circle does not intersect:
b2  4ac  0

A line that touches the circle at one and only one point is called
as tangent to the circle:

b2  4ac  0

The chord of a circle is the one which divides the circle into two
parts called major segment and minor segments:

b2  4ac  0

EXAMPLE 15

Find the intersections between the circle x2  y2  6x  4y  9  0 and the line y  7  x.

y  7  x

x2  y2  6x  4y  9  0

Substitute  into ,

x2  7  x2  6x  47  x  9  0

x2  49  14x  x2  6x  28  4x  9  0
x2  8x  15  0

x  3x  5  0

x  3, x  5

When x  3, When x  5,

y 73 y 75
4 2

 the intersections point are 3, 4 and 5,2.

19 of 42

Lecture Note: 4 of 4 Mathematics 2 SM025
Ch. 4 Conics Session 2021/2022

EXAMPLE 16

The equation of the circle is given by 9x2  y2   18x  6y  26  0.

(a) Find the coordinates of its centre and radius.

(b) Determine whether the point 0, 4  lies inside or outside to the circle.
3

(a) 9x2  y2   18x  6y  26  0

x2  y2  2x  2 y  26  0
39

By comparing;

2g  2, 2f   2 , c   26
g 1 3 9

f 1
3

C g,f   C 1, 1 
3

r 12   1 2   296
3

2

(b) Distance from the point 0, 4  to the center, d  1  02  13  4 2
3 3

2
Since d  r, therefore the point 0, 34 lies inside the circle.

20 of 42

Lecture Note: 4 of 4 Mathematics 2 SM025
Ch. 4 Conics Session 2021/2022

The Equation of Tangents and Normal to a Circle

Method 1: Formula Method 2: Properties of gradient

The diagram shows the tangent to the circle

at point x1, y1 with centre g,f .

g,f 

x1, y1  mN  mT  1

xx1  yy1  g x  x1   f y  y1  c  0 Equation of the tangent is

y  y1  mT x  x1  or y  mT x  c.

If the centre is 0,0, then the equation of

tangent is xx1  yy1  r2.

THE USE OF DERIVATIVE IS NOT ALLOWED!

EXAMPLE 17
Find the equation of the tangent and normal to a circle x2  y2  6x  10y  82  0 at the point

S 1,5.

By comparing;
2g  6, 2f  10, c  82

g  3 f  5

C g,f   C 3,5

mN  5  5
1  3

5
2

Equation of normal;

y  5  5 x  1

2
y  5x5

22

21 of 42

Lecture Note: 4 of 4 Mathematics 2 SM025
Ch. 4 Conics Session 2021/2022

Equation of tangent;

mT mN  1

mT 52  1

mT  2
5

y  5   2 x  1

5
y   2 x  27

55

The Length of a tangent from a Point to a Circle

 d = Length of tangent from point P to circle.
 Using Pythagoras Theorem to find d.

EXAMPLE 18

Find the length of the tangent from point 8,4 to the circle with centre 3,0 and radius 2 units.

Length of BC  8  32  4  02 d

 41

 d  41 2  22

 37

22 of 42

Tutorial 4: 6 Hours Mathematics 2 SM025
Ch. 4 Conics Session 2021/2022

1. Find the equation of a parabola with vertex 2,1, passes through the point 1,7 and has

a horizontal symmetric axis. Sketch the graph of the parabola.

y  12  4p x  2
At point 1,7,
7  12  4p 1  2

36  12p
p3

y  12  43x  2
y  12  12x  2

2. Find the equation of the parabola with its symmetric axis is parallel to the x-axis, vertex

at 2,1 and passes through 6,3. Hence, sketch the graph.

y  12  4p x  2 y

At point 6,3, O x

3  12  4p 6  2 F 3,1

16  16p
p  1

y  12  41x  2
y  12  4x  2

23 of 42

Tutorial 4: 6 Hours Mathematics 2 SM025
Ch. 4 Conics Session 2021/2022

3. An equation x2  4x  4y  8  0 represents a parabola.

(a) Determine the vertex, focus and directrix of the parabola.

(b) Show that the tangent lines to the parabola at the points A2,5 and B 3, 54

intersect at the right angle.

(a) x2  4x  4y  8  0

x  22  4  4y  8  0
x  22  4y  1

4p  4
p 1

F 2,2, V 2,1

Directrix: y  0

x2  4x  4y  8  0
y  1 x2  x  2
4

dy  1 x  1
dx 2

At point 2,5,

mA  dy  1 2  1  2
dx
(b) x2 2

At point 3, 5 ,
4

mB  dy  1 3  1  1
dx 2
x3 2

mAmB  212  1

Since mAmB  1, therefore the tangent lines intersect at the right angle.

24 of 42

Tutorial 4: 6 Hours Mathematics 2 SM025
Ch. 4 Conics Session 2021/2022

4. If the vertex of the parabola y  ax2  bx  2a  b is 1,2, find the values of a and b.

y  ax2  bx  2a  b

ax2  bx  y  2a  b

a x2  bx   y  2a  b
a

a x  b 2   b 2   y  2a  b
2a 2a 

a x  b 2  a  b 2  y  2a b
2a 2a

a x  b 2  y  2a b  a  b 2
2a 2a

x  b  2 1 y 2a  b 2 
2a a 2a
    b  a

By comparing,

 b 1
2a
b  2a

2a  b  a  b 2  2
2a

Substitute  into ,

2a  2a  a22aa2  2
a  2

When a  2, b  22

4

25 of 42

Tutorial 4: 6 Hours Mathematics 2 SM025
Ch. 4 Conics Session 2021/2022

5. Show that the equation y2  9x2  18x  6y  0 represents an ellipse, and sketch the graph.

y2  9x2  18x  6y  0

y  32  9  9 x  12  1  0
y  32  9x  12  18
x  12 y  32

 1
2 18
 the equation y2  9x2  18x  6y  0 represent an ellipse.

C 1,3

c2  b2  a2
 18  2
 16

   V 1,3  18 , V 1,3  18

F 1,1 , F 1,7

0

26 of 42

Tutorial 4: 6 Hours Mathematics 2 SM025
Ch. 4 Conics Session 2021/2022

6. If the ellipse 16x2  y2  px  qy  r  0 touches the x-axis at the point 1,0 and passes

through the point 2,4.

(a) Determine the values of p, q and r.
(b) Express the equation of the ellipse in the standard form.
(c) Find the centre, major vertices and foci of the ellipse.

(a) At point 1,0;

1612  02  p 1  q 0  r  0

16  p  r  0
p  r  16

At point 2,4;
1622  42  p2  q 4  r  0

2p  4q  r  80
32x  2y dy  p  q dy  0

dx dx

dy  0 at 1,0;

dx

321  200  p  q 0  0

p  32
When p  32, 32  r  16

r  16

When p  32, r  16, 232  4q  16  80

q  8

(b) 16x2  32x  y2  8y  16
16(x2  2x)  y2  8y  16

16 x  12  1  y  42  16  16
16(x  1)2  y  42  16
(x  1)2  y  42  1

1 16
(c) a2  1, b2  16, c2  b2  a2

a  1 b  4 c  16  1

c  15

C h,k  C 1,4

V h,k  b  V 1,4  4  V 1,0,V 1,8

F h,k  c  F 1,4  15  F 1,4  15,F 1,4  15

27 of 42

Tutorial 4: 6 Hours Mathematics 2 SM025
Ch. 4 Conics Session 2021/2022

7. (a) Find the equation in standard form of an ellipse which passes through the point 1,6

and having foci at 5,2 and 3,2.

(b) From the result obtained in part (a), sketch the graph of the ellipse.

(a) C 52 3 , 2  2  C 1, 2
2

2c  3  5

2c  8
c4

c2  a2  b2
a2  b2  16

At 1,6,

1  12  6  22 1
b2
a2

16  1
b2

b2  16

Substitute  into ,

a2  16  16
a2  32

 x  12  y  22  1

32 16

(b)
0

28 of 42

Tutorial 4: 6 Hours Mathematics 2 SM025
Ch. 4 Conics Session 2021/2022

8. If 9x2  25y2  225 and k2  1x2  2k2  1y2  k2  12k2  1 has the same foci. Find

the values of k.

9x2  25y2  225
9x2  25y2  225
225 225 225

x2  y2  1
25 9

a2  25, b2  9;

c2  a2  b2

 25  9

 16

k2  1x2  2k2  1y2  k2  12k2  1

k2  1x2 2k2  1y2 k2  12k2  1
k2  12k2  1  k2  12k2  1  k2  12k2  1

x2 1  y2 1  1
2k2  k2 

2k2  1  k2  1  16

k2  16
k  4

9. The length of the major axis of an ellipse is 12. Given the focus is at 0,3 and the minor

axis lies on the line x  2, find its equation.

2a  12, c  2
a6

c2  a2  b2
22  62  b2
b2  32

 x  22  y  32  1

36 32

29 of 42

Tutorial 4: 6 Hours Mathematics 2 SM025
Ch. 4 Conics Session 2021/2022

10. Write the general equation of each of the circle that satisfies the stated conditions.
(a) Tangent to the x-axis, a radius of length 4 units and x-ordinate of center is 3.
(b) Tangent to both axes, a radius of 6 units and the center in the third quadrant.

C 3,4

C 3,4

(a)

Circle with centre, C 3,4,

x  32  y  42  42

x2  y2  6x  8y  9  0

Circle with centre, C 3,4,

x  32  y  42  42

x2  y2  6x  8y  9  0
(b)

C 6,6
x  62  y  62  62

x2  y2  12x  12y  36  0

30 of 42

Tutorial 4: 6 Hours Mathematics 2 SM025
Ch. 4 Conics Session 2021/2022

11. Find the equation of a circle that is passing through points 1,2, 1,2 and 0,1. Hence,

determine its center.

At point 1,2,

12  22  2g 1  2f 2  c  0

c  5  2g  4f 

At point 1,2,

12  22  2g 1  2f 2  c  0

c  5  2g  4f 

At point 0,1,

02  12  2g 0  2f 1  c  0

c  2f  1
Substitute  into ,

5  2g  4f  5  2g  4f
g0

When g  0, c  5  20  4f

 5  4f 

Substitute  into ,

2f  1  5  4f
f 2
3

When f   2 , c  5  4 32
3  7
3

x2  y2  4 y  7  0
33

3x2  3y2  4y  7  0

C g,f   C 0, 2 
3

31 of 42

Tutorial 4: 6 Hours Mathematics 2 SM025
Ch. 4 Conics Session 2021/2022

12. Find the equation of the circle passing through points 1,1 and 3,2 with diameter

y  3x  7  0.

y  3x  7  0

3, 2
1,1 C g,f 

At point1,1,

12  12  2g 1  2f 1  c  0

1  1  2g  2f  c  0
c  2g  2f  2

At point 3,2,

32  22  2g 3  2f 2  c  0

9  4  6g  4f  c  0
c  6g  4f  13

At centre g,f ,

f  3g  7  0

f  3g  7

Substitute  into , Substitute  into , Substitute g   5 into ,
2
2g  2f  2  6g  4f  13 4g  23g  7  11
4g  2f  11 f  3 52  7
g 5  1
2 2

Substitute f   1 , g   5 into ,
22

c  2 52  2 12  2
4

x2  y2  2  5  x  2  12  y  4  0
2

x2  y2  5x  y  4  0

32 of 42

Tutorial 4: 6 Hours Mathematics 2 SM025
Ch. 4 Conics Session 2021/2022

13. A circle which lies in the first quadrant, touches the x-axis, y-axis and the straight line
4x  3y  30  0. Find the equation of the circle.

4x  3y  30  0

r  4r  3r  30

42  32

5r  r  30

r  30  5r or r  30  5r

r   15 r5
2

x  52  y  52  25

x2  y2  10x  10y  25  0

14. If the line 4x  3y  k  0 is the tangent to the circle x2  y2  2x  4y  20  0, find the

values of k.

x2  y2  2x  4y  20  0
By comparing;

2g  2, 2f  4, c  20

g  1 f  2

C g,f   C(1,2)

r  12  22  20

5

41  32  k

5

42  32

5 k2 k  2  25
5 k  23

k  2  25
k  2  25 or

k  27

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Tutorial 4: 6 Hours Mathematics 2 SM025
Ch. 4 Conics Session 2021/2022

15. Given that the straight line x  2y  5  0 intersect with another straight line 3x  y  0

at the point P.
(a) Determine the coordinates of P.

(b) A circle touches the straight line 3x  y  0 at point P and its centre lies on the straight

line 2x  y  0. Find the equation of the circle.

(a) x  2y  5  0
3x  y  0
y  3x
Substitute  into ,

x  23x  5  0

x  1

When x  1, y  31

 3

 P 1,3

(b) 3x  y  0

y  3x

 mN  1
3
Equation of normal;

y  3   1 x  1

3
x  3y  10  0

y  2x

Substitute  into ,

x  32x  10  0

x2

When x  2, y  22

 4

 C 2,4

x  22  y  42  r2

1  22  3  42  r2

r2  10

 x  22  y  42  10

Other option (general equation):
x2  4x  4  y2  8y  16  10

 x2  y2  4x  8y  10  0

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Tutorial 4: 6 Hours Mathematics 2 SM025
Ch. 4 Conics Session 2021/2022

16. Find the equation of the tangents to a circle x2  y2  25 from the point A1,5.

B a,b A 1,5

T1

N1
N2

C 0,0

T2

x2  y 2 52

 C 0,0

mT1  5b, mN1  b0  b
1a a0 a

Since T1 and N1 are perpendicular, therefore mT1  mN1  1, i.e.

15  b  ab   1
 a

5b  b2  1
a  a2

5b  b2  a2  a

5b  a  a2  b2 

At point a,b;

a2  b2  25
Substitute  into ,

5b  a  25

a  25  5b 
Substitute  into ,

25  5b2  b2  25

625  250b  25b2  b2  25  0

13b2  125b  300  0

b  513b  60  0

b  5, b  60
13

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Tutorial 4: 6 Hours Mathematics 2 SM025
Ch. 4 Conics Session 2021/2022

When b  5, a  25  55

0

When b  60 , a  25  51630
13

 25
13

 B 0,5, B 1235 , 60 
13

At point B 0,5;

mT1  55
10

0

y  5  0x  1

y5

At point B 1235 , 60 ;
13

5  60
13
mT2  1  25

13

 5
12

y  5   5 x  1

12

y   5 x  65
12 12

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Tutorial 4: 6 Hours Mathematics 2 SM025
Ch. 4 Conics Session 2021/2022

17. The equation of circle P is given by x2  y2  4x  6y  12  0.
(a) Find the coordinates of its centre and radius.

(b) Find the perpendicular distance from the centre of P to the line 3x  4y  k in term
of k, where k is a constant.

(c) Hence, find the values of k such that the line 3x  4y  k is a tangent to P.

(a) By comparing x2  y2  2gx  2fy  c  0 and x2  y2  4x  6y  12  0,
2g  4, 2f  6, c  12
g  2 f  3

C 2,3

r  22  32  12

5

32  43  k

(b) d 
32  42

 6  k
5

(c) 6  k  5
5

6  k  25

6  k  25, 6  k  25

k  19 k  31

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Tutorial 4: 6 Hours Mathematics 2 SM025
Ch. 4 Conics Session 2021/2022

18. Find the radius and the coordinates of the centre of the circle x2  y2  2x  6y  6  0.

Prove by calculation that the point R 5,6 lies outside the circle. Hence, given that a

tangent to the circle with gradient 4 passes through the point R 5,6, find the equation of

the tangent and its length.

By comparing x2  y2  2gx  2fy  c  0 and x2  y2  2x  6y  6  0,
2g  2, 2f  6, c  6

g  1 f  3

r  12  32  6

2

C g,f   C 1,3

CR  5  12  6  32

5

Since CR  r, therefore the point 5,6 lies outside the circle.

Equation of the tangent, y  6  4x  5

y  4x  14

Lenght of the tangent, d2  52  22

d  21

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Tutorial 4: 6 Hours Mathematics 2 SM025
Ch. 4 Conics Session 2021/2022

19. Let L be a line passing through the centre of the circle x2  y2  2x  2y  7 and
perpendicular to the line 3x  4y  7. Find

(a) the coordinates of the points of intersections of L and the circle.

(b) the equations of the tangents to the circle parallel to 3x  4y  7.

(a) By comparing x2  y2  2x  2y  7  0 and x2  y2  2gx  2fy  c  0,

2g  2, 2f  2, c  7
g  1 f  1

 C 1,1

mK  3
4

 mL  4
3

y  1  4 x  1

3
y  4 x  1 ...

33

x2  y2  2x  2y  7

Substitute  into ,

x2   4x  12  2x  2  4x  1  7  0
3 3

25x2  50x  56  0

5x  45x  14  0

x   4 , x  14
55

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Tutorial 4: 6 Hours Mathematics 2 SM025
Ch. 4 Conics Session 2021/2022

When x   4 , y  4 54  1 When x  14 , y  4154  1
5 3 5 3

 7  17
5 5

 the point of intersections are  4 ,  7  and 154 , 17 .
5 5 5

(b)

Equation of tangents,

y   75   3 x   54
4

or y 3x2
4

y  17   3 x  154 
5 4

y   3 x  11
42

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Tutorial 4: 6 Hours Mathematics 2 SM025
Ch. 4 Conics Session 2021/2022

20. A circle with centre 4,2 touches two perpendicular lines at the points V and W. Given
that the point of intersection between the two lines is 3,3. Find the equation of the

circle.

CV  VP

(4  a)2  2  b2  3  a2  3  b2
(4  a)2  2  b2  3  a2  3  b2

16  8a  a2  4  4b  b2  9  6a  a2  9  6b  b2
2  2a  10b
a  1  5b 

CP  4  32  2  32

 26

CV 2  VP 2  CP 2
4  a2  2  b2  3  a2  3  b2  26

16  8a  a2  4  4b  b2  9  6a  a2  9  6b  b2  26
a2  b2  b  7a  6

Substitute  into ,

(1  5b)2  b2  b  7 1  5b  6

b2  b  0

b b  1  0

b  0, b  1

When b  0, a  1  50 When b  1, a  1  51

1 6

 V 1,0, W 6,1

r2  4  12  2  02

=13

 x  42  y  22  13

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Tutorial 4: 6 Hours Mathematics 2 SM025
Ch. 4 Conics Session 2021/2022

21. Find an equation of the circle passing through the origin and its centre is the focus of the
parabola x2  8y  16.

x2  8y  16

x2  42y  2
 C 0,4
x  02  y  42  42

x2  y2  8y  16  16
x2  y2  8y  0

Answers 11. 3x2  3y2  4y  7  0, C 0, 2 
3
1. y  12  12x  2 , DIY
12. x2  y2  5x  y  4  0
2. y  12  4x  2, DIY
13. x2  y2  10x  10y  25  0
3. (a) F 2,2, V 2,1, y  0 (b) DIY 14. k  27,  23

4. a  2, b  4 15. (a) P 1,3
5. DIY
6. (a) p  32, q  8, r  16 (b) x2  y2  4x  8y  10  0
16. y  5
x  12 y  42
y   5 x  65 (b) d  6  k
(b)   1 12 12 5
1 16
17. (a) C 2,3, r  5
(c) C 1, 4, V 1,0, V 1,8,
(c) k  19,  31
F 1, 4  15, F 1,4  15
18. C 1,3, r  2
7. (a) x  12  y  22  1 (b) DIY
y  4x  14, d  21
32 16
8. k  4 19. (a)  4 , 75  , 154 , 17 
5 5
9. x  22  y  32  1
(b) y   3 x  2 , y   3 x  11
36 32 4 42
10. (a) x2  y2  6x  8y  9  0
20. x  42  y  22  13
x2  y2  6x  8y  9  0
(b) x2  y2  12x  12y  36  0

21. x2  y2  8y  0

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