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

Lines, ratios, circles and tangents

About 90 minutesNo calculator122 marks

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Date

Do Now

The first is what this chapter needs. The rest bring back earlier work so nothing is forgotten.

A1
Needed today
Write $x^{2} - 14 x + 38$ in the form $(x - p)^{2} - q$ .
[2 marks]
Answer

A2
From chapter 3
A line passes through $(- 3, 14)$ and $(2, - 6)$ . Work out its equation in the form $y = m x + c$ .
[2 marks]
Answer $\dots\dots\dots\dots\dots\dots$
A3
From chapter 4
Solve $4 x^{2} - 4 x - 15 = 0$ .
[2 marks]
Answer $x = \dots\dots$ or $x = \dots\dots$
A4
From chapter 1
Simplify $98 + 50$ .
[2 marks]
Answer
A5
GCSE Higher
Expand and simplify $(x - 3)^{2} + (x + 1)^{2}$ .
[2 marks]
Answer

Examples, then your turn

Level 2

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

Example 1

$A$ is $(- 2, 3)$ , $B$ is $(k, 1)$ and $C$ is $(6, 5)$ . Angle $A B C = 9 0^{\circ}$ .

Work out the two possible values of $k$ . Give your answers in surd form.

Not drawn accurately

Hint

A B

and

B C

are perpendicular, so their gradients multiply to

- 1

B1
$P$ is $(- 1, 4)$ , $Q$ is $(3, k)$ and $R$ is $(9, 2)$ . Angle $P QR = 9 0^{\circ}$ .
Work out the two possible values of $k$ .
[4 marks]
Not drawn accurately
Answer $k = \dots$ or $k = \dots$

Example 2

$P$ is the point $(9, - 7)$ . The line $L$ has equation $4 x - 3 y = 7$ .

Work out the coordinates of the foot of the perpendicular from $P$ to $L$ , and the shortest distance from $P$ to $L$ .

Not drawn accurately

Hint The line through

P

perpendicular to

L

meets

L

at the foot

F

. The shortest distance is

P F

B2
$P$ is the point $(3, 7)$ . The line $L$ has equation $x + 3 y = 14$ .
Work out the coordinates of the foot of the perpendicular from $P$ to $L$ . Hence work out the shortest distance from $P$ to $L$ , as a surd.
[5 marks]
Not drawn accurately
Answer $F$ $(\dots, \dots)$ distance $\dots$

Example 3

A circle passes through the points $A (3, 8)$ , $B (9, 0)$ and $C (- 3, - 4)$ . Work out the equation of the circle.

Not drawn accurately

Hint The centre is on the perpendicular bisector of every chord. Find two of them and where they meet.

B3
A circle passes through the points $A (2, 7)$ , $B (4, 1)$ and $C (- 4, - 5)$ . Work out the equation of the circle.
[5 marks]
Not drawn accurately
Answer $\dots\dots\dots\dots\dots\dots$

Example 4

The line $y = 2 x + c$ is a tangent to the circle $x^{2} + y^{2} - 2 x + 4 y - 15 = 0$ . Work out the two possible values of $c$ .

Hint Substitute

y = 2 x + c

into the circle. A tangent meets it once, so the quadratic in

x

has one repeated root.

B4
The line $y = 3 x + c$ is a tangent to the circle $x^{2} + y^{2} + 4 x - 6 y + 3 = 0$ . Work out the two possible values of $c$ . You must show your working.
[5 marks]
Answer $c = \dots$ or $c = \dots$

Practice

Level 2

Each question asks for something different. Show your working.

C1
$P QR S$ is a parallelogram. $P$ is $(- 3, 2)$ , $Q$ is $(4, - 1)$ and $R$ is $(6, 5)$ .
Work out the coordinates of $S$ .
[3 marks]
Not drawn accurately
Answer $(\dots\dots, \dots\dots)$

C2
$A$ is $(- 3, 1)$ , $B$ is $(1, 4)$ and $C$ is $(4, 0)$ .
Show that triangle $A B C$ is right-angled and isosceles. Work out the area of the triangle.
[4 marks]
Not drawn accurately
Answerarea $= \dots\dots$
C3
$A$ is $(- 4, 7)$ and $B$ is $(2, - 1)$ . Work out the equation of the perpendicular bisector of $A B$ . Give your answer in the form $a x + b y = c$ , where $a$ , $b$ and $c$ are integers.
[3 marks]
Not drawn accurately
Answer $\dots\dots\dots\dots\dots\dots$
C4
The lines $y = x + 2$ and $x + 3 y = 26$ and the $x$ -axis enclose a triangle.
Work out the area of the triangle.
[4 marks]
Not drawn accurately
Answer $\dots\dots$ square units
C5
$A$ is $(- 7, 4)$ and $B$ is $(8, - 6)$ . $C$ is the point on $A B$ with $A C : C B = 3 : 2$ .
Work out the coordinates of $C$ .
[2 marks]
Not drawn accurately
Answer $(\dots\dots, \dots\dots)$
C6
A circle has equation $x^{2} + y^{2} - 8 x + 6 y + 5 = 0$ .
Work out the coordinates of its centre and its radius. Give the radius as a surd in its simplest form.
[3 marks]
Answercentre $(\dots, \dots)$ radius $\dots\dots$
C7
A circle has centre $(5, - 4)$ . The $y$ -axis is a tangent to the circle.
Work out the coordinates of the points where the circle crosses the $x$ -axis.
[3 marks]
Not drawn accurately
Answer $(\dots, \dots)$ and $(\dots, \dots)$
C8
Show that the line $y = 2 x + 8$ does not meet the circle $(x - 3)^{2} + (y - 1)^{2} = 16$ .
[3 marks]
C9
The line $x - 2 y = - 5$ meets the circle $x^{2} + y^{2} = 25$ at the points $A$ and $B$ .
Work out the coordinates of $A$ and $B$ .
[4 marks]
Not drawn accurately
Answer $(\dots, \dots)$ and $(\dots, \dots)$

Exam-style questions

AQA exam style

Answer every question in the space given. Show your working: most of the marks are for method.

D1
Circle the centre of the circle $x^{2} + y^{2} - 6 x + 10 y = 2$
$(- 3, 5) (3, - 5) (6, - 10) (- 6, 10)$
[1 mark]

D2
Noah says, "The lines $4 x + 5 y = 7$ and $5 x + 4 y = 2$ are perpendicular, because the numbers in front of $x$ and $y$ have swapped over."
Noah is wrong.
(a)
Show that the two lines are not perpendicular.
[2 marks]
(b)
The line $5 x + k y = 2$ is perpendicular to $4 x + 5 y = 7$ . Work out the value of $k$ .
[2 marks]
Answer $k = \dots\dots$
Total for D2: 4 marks
D3
$A$ is $(- 2, 1)$ and $B$ is $(6, 7)$ . $A B$ is a diameter of a circle.
Not drawn accurately
(a)
Show that the equation of the circle is $x^{2} + y^{2} - 4 x - 8 y - 5 = 0$ .
[3 marks]
(b)
$P (5, 8)$ lies on the circle. Use gradients to show that angle $A P B = 9 0^{\circ}$ .
[2 marks]
(c)
Work out the equation of the tangent to the circle at $P$ . Give your answer in the form $a x + b y = c$ , where $a$ , $b$ and $c$ are integers.
[3 marks]
Answer $\dots\dots\dots\dots\dots\dots$
Total for D3: 8 marks
D4
$A$ is $(- 3, - 5)$ and $B$ is $(9, 4)$ . $P$ is the point on $A B$ with $A P : P B = 1 : 2$ .
Not drawn accurately
(a)
Work out the coordinates of $P$ .
[2 marks]
Answer $(\dots\dots, \dots\dots)$
(b)
A circle has centre $P$ and passes through $A$ . Work out the equation of the circle.
[2 marks]
Answer $\dots\dots\dots\dots\dots\dots$
(c)
The circle crosses the $y$ -axis at two points. Work out the exact distance between them.
[3 marks]
Answer $\dots\dots$
Total for D4: 7 marks
D5
The line $x + 2 y = 20$ meets the $x$ -axis at $P$ and the $y$ -axis at $Q$ .
Not drawn accurately
(a)
Show that the perpendicular bisector of $P Q$ has equation $2 x - y = 15$ .
[3 marks]
(b)
The perpendicular bisector meets the $y$ -axis at $R$ . Work out the area of triangle $P QR$ .
[2 marks]
Answer $\dots\dots$ square units
Total for D5: 5 marks
D6
The line $y = 2 x + 1$ meets the circle $x^{2} + y^{2} + 2 x - 8 y + 7 = 0$ at the points $A$ and $B$ .
Not drawn accurately
(a)
Work out the exact length of $A B$ . You must show your working.
[5 marks]
Answer $\dots\dots$
(b)
Hence work out the shortest distance from the centre of the circle to the line $A B$ .
[2 marks]
Answer $\dots\dots$
Total for D6: 7 marks
D7
$P$ is $(- 2, 7)$ and $Q$ is $(10, - 5)$ . The point $R (k, 3)$ lies on the line $P Q$ .
Not drawn accurately
(a)
Work out the value of $k$ .
[2 marks]
Answer $k = \dots\dots$
(b)
Write down the ratio $P R : R Q$ .
[1 mark]
Answer
(c)
$S$ lies on $P Q$ extended beyond $Q$ , with $P Q : QS = 3 : 1$ . Work out the coordinates of $S$ .
[2 marks]
Answer $(\dots\dots, \dots\dots)$
Total for D7: 5 marks
D8
$A$ is $(1, 2)$ and $B$ is $(6, 7)$ . The point $C$ lies on the line $y = x$ , and $A C = B C$ .
Not drawn accurately
(a)
Work out the coordinates of $C$ . You must show your working.
[4 marks]
Answer $(\dots\dots, \dots\dots)$
(b)
Work out the area of triangle $A B C$ .
[2 marks]
Answer $\dots\dots$ square units
Total for D8: 6 marks
D9
A circle has equation $(x - 2)^{2} + (y - 3)^{2} = 20$ . $T$ is the point $(9, 8)$ .
A tangent is drawn from $T$ to the circle. Work out the length of the tangent. Give your answer as a surd in its simplest form.
[3 marks]
Not drawn accurately
Answer $\dots\dots$

Extension

Stretch

No route is given. Spend five minutes on a problem before you look at a hint. The hints are on the last page of the worksheet, and you can use one at a time.

E1
$A B C D$ is a rhombus. $A$ is $(1, 2)$ and $C$ is $(9, 6)$ . The vertex $B$ lies on the $x$ -axis.
Work out the coordinates of $D$ and the area of the rhombus.
[5 marks]
Not drawn accurately
Hint 1 · What to try
The diagonals of a rhombus bisect each other at right angles. So $B$ and $D$ are on the perpendicular bisector of $A C$ .
Hint 2 · The first line
The perpendicular bisector of $A C$ is $y - 4 = - 2 (x - 5)$ . It meets the $x$ -axis at $B (7, 0)$ .
Hint 3 · The full method
The midpoint $(5, 4)$ is also the midpoint of $B D$ , so $D$ is $(3, 8)$ . Area $= \frac{1}{2} \times A C \times B D = \frac{1}{2} \times 80 \times 80 = 40$ .
Answer $D$ $(\dots, \dots)$ area $\dots\dots$

E2
The line $y = m x$ meets the circle $(x - 5)^{2} + y^{2} = 9$ at two different points.
Work out the range of possible values of $m$ .
[4 marks]
Hint 1 · What to try
Substitute $y = m x$ into the circle and collect the terms in $x$ .
Hint 2 · The first line
$(1 + m^{2}) x^{2} - 10 x + 16 = 0$ . Two different points means two different real roots: $b^{2} - 4 a c > 0$ .
Hint 3 · The full method
$100 - 64 (1 + m^{2}) > 0$ gives $m^{2} < \frac{36}{64}$ , so $- \frac{3}{4} < m < \frac{3}{4}$ .
Answer $\dots\dots\dots\dots$
E3
A circle has equation $x^{2} + y^{2} - 2 x - 4 y - 20 = 0$ . The line $L$ has equation $3 x + 4 y = 51$ .
Work out the shortest distance from the circle to the line $L$ .
[4 marks]
Not drawn accurately
Hint 1 · What to try
Complete the square to find the centre $C$ and the radius. The nearest point of the circle is on the line from $C$ perpendicular to $L$ .
Hint 2 · The first line
The circle is $(x - 1)^{2} + (y - 2)^{2} = 25$ . The line through $C (1, 2)$ perpendicular to $L$ is $4 x - 3 y = - 2$ .
Hint 3 · The full method
That line meets $L$ at $(5.8, 8.4)$ . The distance from $C$ is $4. 8^{2} + 6. 4^{2} = 8$ , so the shortest distance from the circle is $8 - 5 = 3$ .
Answer $\dots\dots$
E4
$A$ is the point $(0, 0)$ and $B$ is the point $(6, 0)$ . The point $P (x, y)$ moves so that $P A = 2 P B$ .
Show that $P$ lies on a circle. Work out its centre and its radius.
[5 marks]
Hint 1 · What to try
Write $P A^{2}$ and $P B^{2}$ in terms of $x$ and $y$ . $P A = 2 P B$ means $P A^{2} = 4 P B^{2}$ .
Hint 2 · The first line
$x^{2} + y^{2} = 4 ((x - 6)^{2} + y^{2})$ , which simplifies to $x^{2} + y^{2} - 16 x + 48 = 0$ .
Hint 3 · The full method
Complete the square: $(x - 8)^{2} + y^{2} = 16$ . That is a circle with centre $(8, 0)$ and radius $4$ .
Answercentre $(\dots, \dots)$ radius $\dots\dots$

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can use gradients, lengths and midpoints to prove facts about shapes, and find an unknown coordinate.
I can find equations of parallel and perpendicular lines, where lines meet, and the shortest distance to a line.
I can find a point that divides a line in a given ratio, inside or beyond the line.
I can find the equation of a circle, and its centre and radius by completing the square.
I can use tangents, chords and the angle in a semicircle, and find where a line meets a circle.

Answers and mark scheme

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