AQA Level 2 Further Maths · Level 2 Certificate · Chapter 5: Coordinate geometry · Lesson 5

Tangents, chords and circles

About 85 minutesNo calculator78 marks

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Date

Do Now

The first questions are what today's lesson needs. The others bring back earlier topics.

A1
Needed today
Work out the gradient of a line perpendicular to the line through $(1, - 2)$ and $(5, 4)$ .
[2 marks]
Answer
A2
Needed today
Solve $x^{2} + (x - 3)^{2} = 29$ .
[2 marks]
Answer $x = \dots\dots$ or $x = \dots\dots$
A3
Last lesson
Work out the centre and the radius of the circle $x^{2} + y^{2} - 4 x + 10 y + 13 = 0$ .
[2 marks]
Answercentre $(\dots, \dots)$ radius $\dots$
A4
From chapter 4
Work out the value of $b^{2} - 4 a c$ for $3 x^{2} - 5 x + 4 = 0$ . How many real solutions does the equation have?
[2 marks]
Answer $\dots\dots$ , $\dots\dots$ solutions

Example, then your turn

Level 2

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

Example 1

$P (7, 1)$ lies on the circle $(x - 3)^{2} + (y + 2)^{2} = 25$ . Work out the equation of the tangent to the circle at $P$ . Give your answer in the form $a x + b y = c$ .

Not drawn accurately

Hint The tangent is perpendicular to the radius

C P

B1
$P (2, 3)$ lies on the circle $(x + 4)^{2} + (y - 1)^{2} = 40$ . Work out the equation of the tangent to the circle at $P$ . Give your answer in the form $y = m x + c$ .
[3 marks]
Answer $\dots\dots\dots\dots\dots\dots$

Example 2

$A (1, - 3)$ and $B (7, 1)$ are points on a circle with centre $C$ . Work out the equation of the line through $C$ that is perpendicular to $A B$ . Give your answer in the form $a x + b y = c$ .

Not drawn accurately

Hint The perpendicular from the centre to a chord passes through the midpoint of the chord.

B2
$A (- 3, 2)$ and $B (5, 6)$ are points on a circle. The centre of the circle lies on the $x$ -axis. Work out the coordinates of the centre.
[4 marks]
Answer $(\dots\dots, \dots\dots)$

Example 3

$A B$ is a diameter of a circle. $A$ is $(- 1, - 2)$ and $B$ is $(7, 4)$ . $P (0, k)$ is a point on the circle, with $k > 0$ . Work out the value of $k$ .

Not drawn accurately

Hint The angle in a semicircle is

9 0^{\circ}

, so

A P

and

B P

are perpendicular: their gradients multiply to

- 1

B3
$A B$ is a diameter of a circle. $A$ is $(- 4, 1)$ and $B$ is $(2, 5)$ . $P (k, 0)$ is a point on the circle, with $k > 0$ . Work out the value of $k$ .
[4 marks]
Answer $k = \dots\dots$

Example 4

Work out the coordinates of the points where the line $y = 2 x - 1$ meets the circle $x^{2} + y^{2} = 13$ .

Hint Put

y = 2 x - 1

into the equation of the circle and solve the quadratic.

B4
Work out the coordinates of the points where the line $x + 2 y = 9$ meets the circle $(x - 2)^{2} + (y - 1)^{2} = 25$ .
[4 marks]
Answer $(\dots, \dots)$ and $(\dots, \dots)$

Example 5

Show that the line $x + y = 6$ is a tangent to the circle $x^{2} + y^{2} = 18$ . Work out the coordinates of the point where it touches the circle.

Hint A line that touches a circle meets it at one point only: the quadratic has one repeated root.

B5
Show that the line $y = 2 x - 5$ is a tangent to the circle $(x - 1)^{2} + (y - 2)^{2} = 5$ . Work out the coordinates of the point where it touches the circle.
[4 marks]
Answer $(\dots\dots, \dots\dots)$

Practice

Level 2

C1
$P (- 2, 6)$ lies on the circle $x^{2} + y^{2} = 40$ . Work out the equation of the tangent to the circle at $P$ . Give your answer in the form $a x + b y = c$ .
[3 marks]
Answer $\dots\dots\dots\dots\dots\dots$
C2
A circle has centre $C (- 3, 2)$ . The chord $P Q$ has midpoint $M (0, 5)$ . Work out the equation of the line $P Q$ .
[3 marks]
Answer $\dots\dots\dots\dots\dots\dots$

C3
$T$ is the point $(9, 5)$ . A tangent is drawn from $T$ to the circle $(x - 1)^{2} + (y + 1)^{2} = 25$ . Work out the length of the tangent. Give your answer as a surd in its simplest form.
[3 marks]
Answer
C4
Show that the line $y = x - 6$ does not meet the circle $x^{2} + y^{2} = 12$ .
[3 marks]
C5
A circle has centre $C (8, k)$ . The line $P T$ is a tangent to the circle at $P (2, 3)$ . $T$ is the point $(0, 7)$ . Work out the value of $k$ .
[3 marks]
Not drawn accurately
Answer $k = \dots\dots$
C6
The line $x + 2 y = 15$ is a tangent to the circle $x^{2} + y^{2} = 45$ at the point $P$ . Work out the coordinates of $P$ .
[3 marks]
Answer $(\dots\dots, \dots\dots)$

Exam-style questions

AQA exam style

D1
Circle the gradient of the tangent to the circle $x^{2} + y^{2} = 29$ at the point $(5, - 2)$ . $- \frac{2}{5} \frac{5}{2} - \frac{5}{2} \frac{2}{5}$
[1 mark]
Answer

D2
Ben says the line $y = 2 x - 5$ is a tangent to the circle $x^{2} + y^{2} = 10$ , because it passes through the point $(3, 1)$ , which is on the circle.
Show that Ben is wrong.
[3 marks]
D3
The circle $(x + 1)^{2} + (y - 3)^{2} = 10$ has centre $C$ . $A (2, 4)$ and $B (- 2, 6)$ are points on the circle. The tangents at $A$ and $B$ meet at $T$ .
Not drawn accurately
(a)
Work out the equation of the tangent at $A$ . Give your answer in the form $y = m x + c$ .
[3 marks]
Answer $\dots\dots\dots\dots\dots\dots$
(b)
The tangent at $B$ has equation $x - 3 y + 20 = 0$ . Work out the coordinates of $T$ .
[2 marks]
Answer $(\dots\dots, \dots\dots)$
(c)
Work out the area of the quadrilateral $C A T B$ .
[3 marks]
Answer $\dots\dots$ units²
Total for D3: 8 marks
D4
The line $y = x + 1$ meets the circle $(x - 3)^{2} + (y - 2)^{2} = 20$ at the points $P$ and $Q$ . Work out the length of $P Q$ . Give your answer as a surd in its simplest form. You must show your working.
[5 marks]
Answer
D5
The circle $(x - 4)^{2} + (y - 7)^{2} = 13$ has centre $C$ . $P (6, 4)$ is a point on the circle. $O$ is the origin.
(a)
Show that $O P$ is a tangent to the circle.
[3 marks]
(b)
Work out the length of $O P$ . Give your answer as a surd in its simplest form.
[2 marks]
Answer
Total for D5: 5 marks
D6
The line $y = x + c$ is a tangent to the circle $(x - 2)^{2} + (y + 1)^{2} = 32$ . Work out the two possible values of $c$ . You must show your working.
[5 marks]
Answer $c = \dots\dots$ or $c = \dots\dots$

Extension

Stretch

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

E1
Two tangents to the circle $(x - 6)^{2} + (y - 2)^{2} = 8$ pass through the origin. Work out the equation of each tangent.
[6 marks]
Hint 1 · What to try
A line through the origin is $y = m x$ . Put it into the equation of the circle.
Hint 2 · The first line
$(1 + m^{2}) x^{2} - (12 + 4 m) x + 32 = 0$ . The line is a tangent when this has one repeated root, so $b^{2} - 4 a c = 0$ .
Hint 3 · The full method
$(12 + 4 m)^{2} - 128 (1 + m^{2}) = 0$ gives $7 m^{2} - 6 m - 1 = 0$ , so $(7 m + 1) (m - 1) = 0$ : $m = 1$ or $m = - \frac{1}{7}$ . The tangents are $y = x$ and $y = - \frac{1}{7} x$ .
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 the equation of the tangent to a circle at a point on it.
I can use "the perpendicular from the centre bisects a chord" and "the angle in a semicircle is 90°" with coordinates.
I can find where a line meets a circle, and show that a line touches it or misses it.
I can work out the length of a tangent from a point outside a circle.

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