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

Equations of straight lines and their intersections

About 85 minutesNo calculator71 marks

Hints online, answers free with an account:
mathswariz.uk/worksheets/further-maths-equations-of-straight-lines-and-their-intersections/

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 the line $5 x + 2 y = 9$ .
[2 marks]
Answer
A2
Needed today
Solve the simultaneous equations $3 x + y = 11$ and $x - 2 y = 6$ .
[2 marks]
Answer $x = \dots\dots y = \dots\dots$
A3
Last lesson
$A$ is $(- 3, 6)$ and $B$ is $(9, 1)$ . Work out the length of $A B$ .
[2 marks]
Answer
A4
From chapter 4
Solve $x^{2} - 3 x - 28 < 0$ .
[2 marks]
Answer

Example, then your turn

Level 2

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

Example 1

Work out the equation of the line that is parallel to $5 x - 2 y + 3 = 0$ and passes through $(2, - 4)$ . Give your answer in the form $a x + b y + c = 0$ .

Hint Parallel lines have the same gradient. Rearrange to

y = \frac{5}{2} x + \frac{3}{2}

, or keep

5 x - 2 y

and change only the number.

B1
Work out the equation of the line that is parallel to $3 x + 4 y - 7 = 0$ and passes through $(- 1, 5)$ . Give your answer in the form $a x + b y + c = 0$ .
[3 marks]
Answer $\dots\dots\dots\dots = 0$

Example 2

Work out the equation of the line that is perpendicular to $3 x - 4 y = 8$ and passes through $(6, - 1)$ . Give your answer in the form $a x + b y = c$ , where $a$ , $b$ and $c$ are integers.

Hint The given line has gradient

\frac{3}{4}

. Perpendicular gradients multiply to

- 1

B2
Work out the equation of the line that is perpendicular to $2 x + 5 y = 9$ and passes through $(- 4, 3)$ . Give your answer in the form $a x + b y = c$ , where $a$ , $b$ and $c$ are integers.
[3 marks]
Answer $\dots\dots\dots\dots$

Example 3

$A$ is $(- 2, 3)$ and $B$ is $(6, 7)$ . Work out the equation of the perpendicular bisector of $A B$ . Give your answer in the form $y = m x + c$ .

Hint The perpendicular bisector goes through the midpoint of

A B

at right angles to

A B

B3
$P$ is $(- 3, - 4)$ and $Q$ is $(5, 2)$ . Work out the equation of the perpendicular bisector of $P Q$ . Give your answer in the form $a x + b y = c$ , where $a$ , $b$ and $c$ are integers.
[4 marks]
Answer $\dots\dots\dots\dots$

Example 4

The line $5 x + 12 y = 60$ meets the $x$ -axis at $P$ and the $y$ -axis at $Q$ . Work out the area of triangle $O P Q$ , the length $P Q$ , and the shortest distance from $O$ to the line.

Not drawn accurately

Hint Put

y = 0

, then

x = 0

. The shortest distance

d

is the height of the triangle with base

P Q

: area

= \frac{1}{2} \times P Q \times d

B4
The line $2 x - 5 y = 20$ meets the $x$ -axis at $A$ and the $y$ -axis at $B$ . Work out the area of triangle $O A B$ , and the equation of the line through $O$ that is perpendicular to $A B$ .
[4 marks]
Not drawn accurately
Answerarea $= \dots\dots$ line: $\dots\dots\dots\dots$

Example 5

The lines $y = 2 x - 1$ , $x + y = 8$ and $y = 1$ enclose a triangle. Work out the coordinates of its vertices and its area.

Not drawn accurately

Hint Each vertex is where two of the lines meet: solve each pair simultaneously.

B5
The lines $y = 3 x + 4$ , $2 x + y = 9$ and $y = - 2$ enclose a triangle. Work out the coordinates of its vertices and its area.
[5 marks]
Not drawn accurately
Answerarea $= \dots\dots$

Practice

Level 2

C1
For each pair, state whether the lines are parallel, perpendicular or neither. Show how you decide. (a) $y = 3 x - 2$ and $6 x - 2 y = 5$ (b) $2 x + 5 y = 1$ and $5 x - 2 y = 4$ (c) $x + 4 y = 3$ and $4 x + y = 3$
[3 marks]
Answer
C2
The lines $3 x + k y = 4$ and $4 x - 6 y = 1$ are perpendicular. Work out the value of $k$ .
[3 marks]
Answer $k = \dots\dots$

C3
Triangle $A B C$ has vertices $A (1, 6)$ , $B (- 3, - 2)$ and $C (7, 0)$ . A median joins a vertex to the midpoint of the opposite side. Work out the equation of the median from $A$ .
[3 marks]
Not drawn accurately
Answer $\dots\dots\dots\dots$
C4
The graph shows the lines $x + 2 y = 7$ and $y = x - 2$ . (a) Use the graph to estimate the solution of the simultaneous equations $x + 2 y = 7$ and $y = x - 2$ . (b) Solve the equations algebraically to find the exact solution.
[3 marks]
Answer(a) $(\dots, \dots)$ (b) $x = \dots\dots y = \dots\dots$

Exam-style questions

AQA exam style

D1
Circle the gradient of a line that is perpendicular to $4 x - 3 y = 12$ . $- \frac{4}{3} - \frac{3}{4} \frac{3}{4} \frac{4}{3}$
[1 mark]
Answer

D2
Ella wants the line through $(4, - 2)$ that is perpendicular to $y = 2 x + 5$ . She writes $y = - 2 x + 6$ .
Explain Ella's mistake. Work out the correct equation.
[3 marks]
Answer $y = \dots\dots\dots$
D3
Taxi firm $A$ charges £ $3.20$ plus £ $1.40$ per mile. Taxi firm $B$ charges £ $5$ plus £ $1.10$ per mile. The cost of a journey of $d$ miles is $C$ pounds.
(a)
Write down an equation for $C$ in terms of $d$ for each firm.
[1 mark]
Answer
(b)
Work out the length of journey for which the two firms charge the same, and that cost. You must show your working.
[3 marks]
Answer $\dots\dots$ miles, £ $\dots\dots$
Total for D3: 4 marks
D4
$A$ is $(- 2, 1)$ , $B$ is $(4, 4)$ and $C$ is $(5, - 3)$ . The line through $C$ perpendicular to $A B$ meets $A B$ at $D$ .
Not drawn accurately
(a)
Work out the equation of $A B$ . Give your answer in the form $y = m x + c$ .
[2 marks]
Answer $y = \dots\dots\dots$
(b)
Work out the coordinates of $D$ .
[4 marks]
Answer $(\dots\dots, \dots\dots)$
(c)
Hence work out the area of triangle $A B C$ .
[3 marks]
Answer
Total for D4: 9 marks
D5
Triangle $P QR$ is bounded by the $x$ -axis, the line $y = 2 x - 6$ and a line through $Q (5, 4)$ . $P$ and $R$ are on the $x$ -axis. Angle $P QR = 9 0^{\circ}$ . Work out the area of triangle $P QR$ . You must show your working.
[5 marks]
Not drawn accurately
Answer
D6
The lines $y = 2 x - 3$ and $x + 3 y = 5$ meet at the point $P$ .
(a)
Show that the line $4 x - 3 y = 5$ also passes through $P$ .
[3 marks]
(b)
Hence work out the equation of the line through $P$ that is parallel to $4 x + y = 1$ . Give your answer in the form $a x + b y = c$ .
[2 marks]
Answer $\dots\dots\dots\dots$
Total for D6: 5 marks

Extension

Stretch

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

E1
A straight line passes through the point $(4, 3)$ . It meets the positive $x$ -axis at $A$ and the positive $y$ -axis at $B$ . The area of triangle $O A B$ is $24$ . Work out the equation of the line.
[5 marks]
Hint 1 · What to try
Call the intercepts $A (a, 0)$ and $B (0, b)$ . Write down two equations: one for the area, one saying $(4, 3)$ is on the line.
Hint 2 · The first line
The line is $\frac{x}{a} + \frac{y}{b} = 1$ , so $\frac{4}{a} + \frac{3}{b} = 1$ , and $\frac{1}{2} ab = 24$ .
Hint 3 · The full method
Multiply by $ab$ : $4 b + 3 a = ab = 48$ , so $a = \frac{48 - 4 b}{3}$ . Then $b (48 - 4 b) = 144$ , $b^{2} - 12 b + 36 = 0$ , $b = 6$ and $a = 8$ . The line is $3 x + 4 y = 24$ .
Answer $\dots\dots\dots\dots$

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can write the equation of a line parallel or perpendicular to a given line through a point.
I can find a perpendicular bisector and the foot of a perpendicular.
I can find where lines meet the axes and each other by solving simultaneously.
I can work out areas of triangles made by lines.

Answers and mark scheme

The worked answers and the full mark scheme are free with a Mathswariz account. Teachers and students can both sign up.

Get the answers