GCSE Maths · Higher · Chapter 23: Graphs · Lesson 3

The equation of a circle

Grades 7–9About 45 minutesCalculator allowed44 marks

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NameClassDate

Do Now

Grades 4–6

The first question is what today's lesson needs. The others bring back earlier topics.

A1
Needed today
A right-angled triangle has hypotenuse 15 cm and one side 9 cm. Find the other side.
[1 mark]
Answer
A2
From chapter 15
Solve $3 x + y = 16$ and $x + y = 6$ .
[1 mark]
Answer
A3
From chapter 21
$y \propto x^{2}$ , and $y = 45$ when $x = 3$ . Find $y$ when $x = 7$ .
[1 mark]
Answer
A4
From chapter 22
You know two angles and a side; you want another side. Which rule do you use?
[1 mark]
Answer

Grades 7–8

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

Example 1

A circle has centre $O$ and passes through $(6, - 11)$ . Find its equation and its radius, to 1 d.p.

Hint

r^{2}

is the square of the distance from

O

to the point.

B1
A circle has centre $O$ and passes through $(- 8, 13)$ . Find its equation and its radius, to 1 d.p.
[3 marks]
Answer

Example 2

$P (4, 8)$ lies on the circle $x^{2} + y^{2} = 80$ . Find the equation of the tangent at $P$ .

Not drawn accurately

Hint The tangent is perpendicular to the radius

O P

B2
$P (8, 2)$ lies on the circle $x^{2} + y^{2} = 68$ . Find the equation of the tangent at $P$ .
[3 marks]
Not drawn accurately
Answer

Grades 7–8

C1
Is the point $(7, - 8)$ inside, on or outside the circle $x^{2} + y^{2} = 110$ ? Show how you decide.
[2 marks]
Answer
C2
The point $(p, 9)$ lies on the circle $x^{2} + y^{2} = 130$ , and $p$ is negative. Find $p$ .
[2 marks]
Answer

C3
Find the radius of the circle $x^{2} + y^{2} = 63$ . Give your answer in the form $a b$ .
[2 marks]
Answer
C4
The point $(k, 3 k)$ lies on the circle $x^{2} + y^{2} = 160$ , and $k > 0$ . Find $k$ .
[2 marks]
Answer
C5
How many points with whole-number coordinates lie on the circle $x^{2} + y^{2} = 53$ ?
[2 marks]
Answer
C6
$A (- 5, 8)$ and $B (5, - 8)$ are the ends of a diameter of a circle. Find the equation of the circle.
[2 marks]
Answer
C7
$T (12, 5)$ is outside the circle $x^{2} + y^{2} = 58$ . A tangent from $T$ touches the circle at $P$ . Find $T P$ , to 1 d.p.
[3 marks]
Not drawn accurately
Answer
C8
Find the exact circumference of the circle $x^{2} + y^{2} = 48$ . Give your answer in the form $k 3 π$ .
[2 marks]
Answer

Grades 8–9

D1
$P (8, 9)$ lies on the circle $x^{2} + y^{2} = 145$ . Dan says, "The tangent to the circle at $P$ has gradient $\frac{9}{8}$ ."
Explain why Dan is wrong. Find the equation of the tangent in the form $a x + b y = c$ , where $a$ , $b$ and $c$ are integers.
[3 marks]
Answer

D2
The point $P (12, 6)$ and the circle $x^{2} + y^{2} = 180$ . The tangent at $P$ meets the $x$ -axis at $A$ .
Not drawn accurately
(a)
Show that $P$ lies on the circle.
[1 mark]
Answer
(b)
Find the equation of the tangent at $P$ , in the form $y = m x + c$ .
[3 marks]
Answer
(c)
Work out the area of triangle $O P A$ .
[2 marks]
Answer
Total for D2: 6 marks
D3
The line $L$ has gradient $- 3$ and is a tangent to the circle $x^{2} + y^{2} = 250$ . Find the two possible equations of $L$ .
[4 marks]
Answer

Grade 9 and beyond

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

Colour one circle on each line.

	Not yet	Nearly	Yes
I can use $x^{2} + y^{2} = r^{2}$ .
I can tell if a point is inside or outside.
I can find a tangent to a circle.

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