A-Level Maths (Edexcel 9MA0) · Pure Year 1 · Chapter 6: Circles · Lesson 3

Tangents, chords and circles through triangles

74 marks

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Date

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Quick questions to start.

A1
Needed today
The line $l$ passes through $(1, - 2)$ and $(4, 4)$ . Find the gradient of a line perpendicular to $l$ .
[2 marks]
Answer
A2
Needed today
Find the equation of the line through $(3, - 1)$ with gradient $\frac{2}{3}$ , in the form $a x + b y + c = 0$ where $a$ , $b$ and $c$ are integers.
[2 marks]
Answer
A3
Last lesson
Show that the line $y = x + 4$ is a tangent to the circle $x^{2} + y^{2} = 8$ , and find the point where it touches.
[2 marks]
Answer $(\dots\dots, \dots\dots)$
A4
GCSE Higher
$P$ is $(1, 2)$ , $Q$ is $(3, 6)$ and $R$ is $(7, 4)$ . Work out $P Q^{2}$ , $Q R^{2}$ and $P R^{2}$ . What do they tell you about angle $P QR$ ?
[2 marks]
Answer

Example, then your turn

A-Level

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

Example 1

The point $P (7, 5)$ lies on the circle $(x - 3)^{2} + (y - 2)^{2} = 25$ . Find the equation of the tangent to the circle at $P$ , in the form $a x + b y + c = 0$ .

Hint The tangent is perpendicular to the radius

C P

. Find the gradient of

C P

first.

B1
The circle $C$ has equation $x^{2} + y^{2} - 4 x + 8 y - 5 = 0$ . Show that the point $P (5, 0)$ lies on $C$ , and find the equation of the tangent to $C$ at $P$ in the form $a x + b y + c = 0$ .
[4 marks]
Answer $\dots\dots\dots\dots = 0$

Example 2

Find the equations of the two tangents to the circle $(x - 4)^{2} + (y - 1)^{2} = 18$ that have gradient $1$ .

Hint At each point of contact the radius is perpendicular to the tangent, so the radius has gradient

- 1

. Go out from the centre along a radius of gradient

- 1

B2
Find the equations of the two tangents to the circle $(x + 2)^{2} + (y - 3)^{2} = 45$ that have gradient $\frac{1}{2}$ .
[5 marks]
Answer $y = \dots\dots$ and $y = \dots\dots$

Example 3

The triangle $A B C$ has vertices $A (- 3, 3)$ , $B (5, 7)$ and $C (5, - 1)$ . Find the equation of the circle that passes through $A$ , $B$ and $C$ .

Hint The centre is on the perpendicular bisector of

A B

AND on the perpendicular bisector of

B C

. Find where they cross.

B3
The triangle $P QR$ has vertices $P (- 4, 3)$ , $Q (2, 5)$ and $R (8, - 3)$ . Find the equation of the circle that passes through $P$ , $Q$ and $R$ .
[6 marks]
Answer

Practice

A-Level

C1
The points $A (- 3, 2)$ , $B (1, 6)$ and $C (7, 0)$ lie on a circle. Show that $A C$ is a diameter of the circle, and find the equation of the circle.
[5 marks]
Answer

C2
$A (4, 7)$ and $B (- 2, - 1)$ are the ends of a diameter of a circle. The point $C$ lies on the positive $x$ -axis and angle $A C B = 9 0^{\circ}$ . Find the coordinates of $C$ .
[5 marks]
Answer $(\dots\dots, \dots\dots)$
C3
The circle $x^{2} + y^{2} - 6 x - 2 y + 5 = 0$ has centre $C$ . The point $T$ is $(8, 6)$ . A tangent from $T$ touches the circle at $P$ . Find the exact length of $T P$ .
[5 marks]
Not drawn accurately
Answer $T P = \dots\dots$

Exam-style questions

Edexcel exam style

D1
The points $A (1, - 1)$ , $B (- 3, 1)$ and $C (5, q)$ lie on a circle, where $q$ is a constant. Angle $B A C = 9 0^{\circ}$ .
Find the value of $q$ , and the equation of the circle.
[6 marks]
Answer $q = \dots\dots$ ; $\dots\dots\dots\dots$

D2
The circle $C$ has equation $(x + 1)^{2} + (y - 4)^{2} = 20$ . The point $P (3, 6)$ lies on $C$ . Sam finds the tangent to $C$ at $P$ . He writes:
The gradient from $(- 1, 4)$ to $(3, 6)$ is $\frac{1}{2}$ , so the tangent is $y - 6 = \frac{1}{2} (x - 3)$ .
Sam is wrong.
(a)
Explain the mistake Sam has made.
[1 mark]
(b)
Find the equation of the tangent to $C$ at $P$ .
[3 marks]
Answer
(c)
The tangent meets the $x$ -axis at $R$ . Find the coordinates of $R$ . Hence find the area of triangle $C P R$ , where $C$ is the centre of the circle.
[4 marks]
Answer $R (\dots\dots, \dots\dots)$ , area $\dots\dots$
Total for D2: 8 marks
D3
The circle $C$ has equation $x^{2} + y^{2} - 4 x - 6 y + 8 = 0$ . The points $P (3, 5)$ and $Q (3, 1)$ lie on $C$ . The tangents to $C$ at $P$ and at $Q$ meet at the point $R$ .
Not drawn accurately
(a)
Find the centre and the radius of $C$ .
[2 marks]
Answer
(b)
Find the equation of the tangent to $C$ at $P$ , in the form $a x + b y + c = 0$ .
[3 marks]
Answer
(c)
The tangent at $Q$ has equation $x - 2 y - 1 = 0$ . Find the coordinates of $R$ . Hence find the area of the quadrilateral $C P R Q$ , where $C$ is the centre of the circle.
[4 marks]
Answer $R (\dots\dots, \dots\dots)$ , area $\dots\dots$
Total for D3: 9 marks
D4
A circle passes through the points $A (0, 4)$ and $B (6, 0)$ . The centre of the circle lies on the line $y = x + 1$ .
Not drawn accurately
(a)
Find the equation of the perpendicular bisector of $A B$ , in the form $a x + b y + c = 0$ .
[3 marks]
Answer
(b)
Hence find the equation of the circle. The circle meets the $y$ -axis at $A$ and at a second point $D$ . Find the coordinates of $D$ .
[4 marks]
Answer $\dots\dots\dots\dots$ ; $D (\dots\dots, \dots\dots)$
Total for D4: 7 marks

Extension

Stretch

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

E1
The point $T (7, 1)$ lies outside the circle $x^{2} + y^{2} = 25$ . The two tangents from $T$ touch the circle at $P$ and $Q$ . Find the coordinates of $P$ and $Q$ , and show that the two tangents are perpendicular to each other.
[6 marks]
Hint 1 · What to try
Angle $O P T$ is $9 0^{\circ}$ , so $P$ lies on the circle that has $O T$ as a diameter. So does $Q$ .
Hint 2 · The first line
The circle with diameter $O T$ is $x^{2} + y^{2} - 7 x - y = 0$ . Subtract it from $x^{2} + y^{2} = 25$ to get a straight line through $P$ and $Q$ .
Hint 3 · The full method
$7 x + y = 25$ , so $y = 25 - 7 x$ and $x^{2} - 7 x + 12 = 0$ : $P (3, 4)$ and $Q (4, - 3)$ . The tangents are $3 x + 4 y = 25$ and $4 x - 3 y = 25$ , with gradients $- \frac{3}{4}$ and $\frac{4}{3}$ , whose product is $- 1$ .
Answer $(\dots\dots, \dots\dots)$ and $(\dots\dots, \dots\dots)$

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can find a tangent to a circle using the fact that it is perpendicular to the radius.
I can use the perpendicular bisector of a chord to find the centre of a circle.
I can find the circle through three points, and spot when one side is a diameter.
I can work out lengths and areas made by tangents and radii.

Answers and mark scheme

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