A-Level Maths (Edexcel 9MA0) · Pure Year 1 · Chapter 6: Circles · Lesson 2

Where a line meets a circle

75 marks

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Date

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Quick questions to start.

A1
Needed today
Find the discriminant of $3 x^{2} + 5 x + 4$ . How many real roots does $3 x^{2} + 5 x + 4 = 0$ have?
[2 marks]
Answer
A2
Needed today
Substitute $y = 3 x - 2$ into $x^{2} + y^{2} = 10$ . Write the result in the form $a x^{2} + b x + c = 0$ .
[2 marks]
Answer
A3
Last lesson
Find the centre and the radius of the circle $x^{2} + y^{2} - 8 x + 6 y - 11 = 0$ .
[2 marks]
Answer
A4
From chapter 5
Find the equation of the line through $(2, - 1)$ perpendicular to $y = 3 x + 4$ , in the form $a x + b y + c = 0$ .
[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 circle $(x - 3)^{2} + (y + 2)^{2} = 20$ meets the $x$ -axis at two points. Find their coordinates.

Hint On the

x

-axis,

y = 0

. Substitute and solve for

x

B1
The circle $(x + 2)^{2} + (y - 4)^{2} = 25$ meets the $y$ -axis at two points. Find their coordinates, giving your answers in surd form.
[3 marks]
Answer

Example 2

Find the coordinates of the points where the line $y = x - 1$ meets the circle $(x - 2)^{2} + (y - 3)^{2} = 10$ .

Hint Replace

y

x - 1

in the circle's equation. You get a quadratic in

x

. Then find each

y

from the LINE.

B2
The line $x + y = 4$ meets the circle $x^{2} + y^{2} - 4 x - 2 y - 20 = 0$ at $A$ and $B$ . Find the coordinates of $A$ and $B$ .
[4 marks]
Answer $(\dots\dots, \dots\dots)$ and $(\dots\dots, \dots\dots)$

Example 3

Show that the line $y = 2 x + 6$ does not meet the circle $(x - 4)^{2} + (y - 1)^{2} = 9$ .

Hint Substitute, simplify to

a x^{2} + b x + c = 0

, then look at the sign of

b^{2} - 4 a c

B3
Show that the line $3 x + y = 15$ does not meet the circle $x^{2} + y^{2} + 2 x - 4 y - 4 = 0$ .
[4 marks]

Example 4

The line $y = x + k$ is a tangent to the circle $(x - 3)^{2} + y^{2} = 8$ . Find the two possible values of $k$ and the point of contact for each.

Hint A tangent meets the circle at ONE point, so the quadratic has a repeated root:

b^{2} - 4 a c = 0

B4
The line $y = 2 x + c$ is a tangent to the circle $(x - 1)^{2} + (y - 2)^{2} = 20$ . Find the two possible values of $c$ .
[5 marks]
Answer $c = \dots\dots$ or $c = \dots\dots$

Practice

A-Level

C1
The line $x - 2 y + 5 = 0$ meets the circle $(x + 1)^{2} + (y - 2)^{2} = 45$ at $A$ and $B$ . Show that $A B$ is a diameter of the circle.
[4 marks]

C2
The line $x - y + 3 = 0$ meets the circle $x^{2} + y^{2} = 65$ at $P$ and $Q$ . Find the exact length of $P Q$ .
[5 marks]
Not drawn accurately
Answer $P Q = \dots\dots$
C3
Show that the line $x + 3 y = 9$ is a tangent to the circle $(x - 2)^{2} + (y + 1)^{2} = 10$ , and find the coordinates of the point where it touches the circle.
[4 marks]
Answer $(\dots\dots, \dots\dots)$

Exam-style questions

Edexcel exam style

D1
Priya is asked whether the line $y = 9 - 2 x$ meets the circle $x^{2} + y^{2} - 2 x - 4 y - 5 = 0$ . She writes:
$x^{2} + 81 + 4 x^{2} - 2 x - 36 + 8 x - 5 = 0$ , so $5 x^{2} + 6 x + 40 = 0$ . $b^{2} - 4 a c = 36 - 800 < 0$ , so the line does not meet the circle.
Priya is wrong.
(a)
Identify the mistake Priya has made.
[1 mark]
(b)
Find the coordinates of the points $A$ and $B$ where the line meets the circle.
[4 marks]
Answer $(\dots\dots, \dots\dots)$ and $(\dots\dots, \dots\dots)$
(c)
Show that the perpendicular bisector of $A B$ passes through the centre of the circle.
[3 marks]
Total for D1: 8 marks

D2
The circle $C$ has equation $x^{2} + y^{2} - 6 x + 2 y + 5 = 0$ . The line $l$ has equation $y = k x - 2$ , where $k$ is a constant.
Not drawn accurately
(a)
Show that the $x$ -coordinates of any points where $l$ meets $C$ satisfy $(1 + k^{2}) x^{2} - (2 k + 6) x + 5 = 0$ .
[2 marks]
(b)
Given that $l$ meets $C$ at two distinct points, show that $2 k^{2} - 3 k - 2 < 0$ . Hence find the range of possible values of $k$ .
[5 marks]
Answer
Total for D2: 7 marks
D3
The circle with equation $(x - 2)^{2} + (y + 1)^{2} = 25$ has centre $C$ . The line $l$ has equation $y = x + 2$ . The line $l$ meets the circle at $P$ and $Q$ , where $P$ is to the left of $Q$ .
Not drawn accurately
(a)
Find the coordinates of $P$ and $Q$ .
[5 marks]
Answer $P (\dots\dots, \dots\dots)$ , $Q (\dots\dots, \dots\dots)$
(b)
Show that angle $P C Q$ is $9 0^{\circ}$ . Hence find the exact area of the smaller region bounded by the circle and the chord $P Q$ .
[6 marks]
Answer
Total for D3: 11 marks
D4
A circular lawn is modelled by the circle $x^{2} + y^{2} - 12 x - 8 y + 27 = 0$ , where the units are metres. A straight path is modelled by the line $y = 2 x - 3$ . The path crosses the edge of the lawn at $A$ and $B$ .
Find the length $A B$ of the path that crosses the lawn, giving your answer in the form $a 5$ metres.
[6 marks]
Not drawn accurately
Answer $\dots\dots$ m

Extension

Stretch

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

E1
The line $y = k x$ meets the circle $x^{2} + y^{2} - 10 x + 16 = 0$ at two distinct points $A$ and $B$ . Find the range of values of $k$ . Show that, as $k$ varies, the midpoint of $A B$ always lies on one fixed circle, and find the centre and radius of that circle.
[6 marks]
Hint 1 · What to try
Write down the quadratic in $x$ . The midpoint's $x$ -coordinate is half the SUM of the roots, which you can read off the quadratic without solving it.
Hint 2 · The first line
$(1 + k^{2}) x^{2} - 10 x + 16 = 0$ . The midpoint is $(\frac{5}{1 + k ^{2}}, \frac{5 k}{1 + k ^{2}})$ .
Hint 3 · The full method
$100 - 64 (1 + k^{2}) > 0$ gives $- \frac{3}{4} < k < \frac{3}{4}$ . The midpoint has $x^{2} + y^{2} = \frac{25}{1 + k ^{2}} = 5 x$ : the circle with centre $(\frac{5}{2}, 0)$ and radius $\frac{5}{2}$ .
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can find where a line meets a circle by substituting and solving a quadratic.
I can use the discriminant to decide if a line crosses, touches or misses a circle.
I can find an unknown constant when a line is a tangent or misses a circle.
I can use the points found to work out lengths, midpoints and areas.

Answers and mark scheme

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