A-Level Maths (Edexcel 9MA0) · Pure Year 1 · Chapter 6: Circles · Lesson 1

Midpoints, perpendicular bisectors and the equation of a circle

74 marks

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Quick questions to start.

A1
GCSE Higher
Find the midpoint and the length of the line segment joining $(- 3, - 6)$ and $(5, 0)$ .
[2 marks]
Answer $(\dots, \dots)$ $\dots\dots$
A2
From chapter 5
Find the equation of the line through $(1, 4)$ perpendicular to $y = 2 x - 3$ , in the form $a x + b y + c = 0$ .
[2 marks]
Answer $\dots\dots = 0$
A3
Needed today
Write $x^{2} - 10 x + 7$ in the form $(x - p)^{2} + q$ .
[2 marks]
Answer $\dots\dots$
A4
From chapter 2
Solve $(k - 3)^{2} + 16 = 25$ .
[2 marks]
Answer $k = \dots\dots$

Example, then your turn

A-Level

Your teacher works through each example with you. Then try the one beside it.

Example 1

$A$ is $(- 3, 2)$ and $B$ is $(7, 6)$ . Find the perpendicular bisector of $A B$ , in the form $a x + b y + c = 0$ .

Hint It passes through the midpoint of

A B

and is perpendicular to

A B

B1
$P$ is $(- 9, - 5)$ and $Q$ is $(- 1, - 1)$ . Find the perpendicular bisector of $P Q$ , in the form $a x + b y + c = 0$ .
[4 marks]
Answer $\dots\dots = 0$

Example 2

A circle has centre $(- 2, 4)$ and passes through $(4, - 4)$ . Find its equation, and decide whether $(5, - 1)$ is inside, on or outside it.

Hint The radius is the distance from the centre to the point.

B2
A circle has centre $(3, - 1)$ and passes through $(8, 11)$ . Find its equation, show that $(15, 4)$ lies on it, and decide whether $(- 6, 3)$ is inside, on or outside it.
[4 marks]
Answer

Example 3

Find the centre and the radius of the circle $x^{2} + y^{2} + 6 x - 10 y - 2 = 0$ .

Hint Complete the square in

x

and in

y

separately.

B3
Find the centre and the radius of the circle $2 x^{2} + 2 y^{2} - 8 x + 12 y - 6 = 0$ .
[4 marks]
Answer $(\dots, \dots)$ , $r = \dots\dots$

Example 4

$x^{2} + y^{2} + 8 x - 2 y + k = 0$ is the equation of a circle. Find the range of possible values of $k$ , and the value of $k$ when the radius is $3$ .

Hint After completing the square, the right-hand side is

r^{2}

, and it must be positive.

B4
A circle has equation $x^{2} + y^{2} - 10 x - 6 y + k = 0$ . Find the range of possible values of $k$ , and the value of $k$ for which the circle touches the $x$ -axis.
[4 marks]
Answer $\dots\dots$ $k = \dots\dots$

Practice

A-Level

C1
The midpoint of $P Q$ is $(2, - 5)$ , and $P$ is $(7, 1)$ . Find the coordinates of $Q$ , and the exact length of $P Q$ .
[3 marks]
Answer $(\dots, \dots)$ $\dots\dots$

C2
$A (- 6, 1)$ and $B (4, - 3)$ are the ends of a diameter of a circle. Find the equation of the circle, and show that $(- 3, 4)$ lies on it.
[4 marks]
Answer $\dots\dots$
C3
The point $(k, 5)$ lies on the circle $(x - 2)^{2} + (y + 1)^{2} = 61$ . Find the two possible values of $k$ .
[3 marks]
Answer $k = \dots\dots$
C4
$A$ is $(- 1, 3)$ and $B$ is $(5, 7)$ . Show that the perpendicular bisector of $A B$ passes through $C (8, - 4)$ .
[4 marks]

Exam-style questions

Edexcel exam style

D1
The circle $C$ has equation $x^{2} + y^{2} + 10 x - 4 y - 7 = 0$ . Ben says: “Halve the $10$ and the $- 4$ : the centre of $C$ is $(5, - 2)$ .”
Ben is wrong.
(a)
Find the centre and the radius of $C$ , and explain Ben's mistake.
[5 marks]
(b)
Show that $A (1, 2)$ lies on $C$ . Hence find the coordinates of $B$ , where $A B$ is a diameter of $C$ .
[3 marks]
Answer $B (\dots, \dots)$
Total for D1: 8 marks

D2
The points $A (- 1, 10)$ and $B (5, 8)$ are shown.
Not drawn accurately
(a)
Find the perpendicular bisector of $A B$ , in the form $a x + b y + c = 0$ .
[4 marks]
Answer $\dots\dots = 0$
(b)
A circle passes through $A$ and $B$ , and its centre lies on the $y$ -axis. Find its equation.
[3 marks]
Answer $\dots\dots$
(c)
Hence find the other point on the circle with $y$ -coordinate $10$ .
[2 marks]
Answer $(\dots, \dots)$
Total for D2: 9 marks
D3
The circle $C$ has equation $x^{2} + y^{2} - 2 k x + 6 y + k = 0$ , where $k$ is a constant.
(a)
Find the centre of $C$ , and show that its radius $r$ satisfies $r^{2} = k^{2} - k + 9$ .
[3 marks]
(b)
Hence show that $C$ is a circle for every real value of $k$ .
[2 marks]
(c)
Given that the radius of $C$ is $11$ , find the two possible values of $k$ .
[3 marks]
Answer $k = \dots\dots$
Total for D3: 8 marks
D4
Circle $C_{1}$ has equation $x^{2} + y^{2} - 4 x - 2 y - 4 = 0$ and circle $C_{2}$ has equation $x^{2} + y^{2} + 8 x + 14 y + 16 = 0$ . Find the centre and radius of each, and hence show that they touch.
[5 marks]

Extension

Stretch

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
A circle touches both axes and passes through $(1, 8)$ . Find its two possible equations.
[6 marks]
Hint 1 · What to try
It touches both axes, so its centre is $(r, r)$ .
Hint 2 · The first line
The distance from $(r, r)$ to $(1, 8)$ is $r$ .
Hint 3 · The full method
$r^{2} - 18 r + 65 = 0$ .
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can bisect a line segment.
I can write a circle equation.
I can find centre and radius.
I can find a missing value.

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