A-level · Coordinate geometry

Straight linesExam-style questions

The gradient of a line as a constant rate of change, the three forms of a line's equation — y = mx + c, y − y₁ = m(x − x₁), and ax + by + c = 0 — and how to move between them.

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10 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 2 Level 3 Level 4 Level 5

1
In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable.Figure 1

Figure 1 shows the straight line $l_1$ , which passes through the points $A(-3,\ 6)$ and $B(7,\ 1)$ .
(a) Find an equation for $l_1$ , giving your answer in the form $y = mx + c$ , where $m$ and $c$ are constants.

(3 marks)

The point $C(0,\ 8)$ lies on the $y$ -axis. The straight line $l_2$ passes through $C$ and is perpendicular to $l_1$ . Figure 1 also shows the point $R$ , where $l_2$ crosses $l_1$
(b) Show that $R$ has coordinates $\left(-\tfrac75,\ \tfrac{26}{5}\right)$

(4 marks)
(c) Find the exact area of triangle $ABC$ .

(3 marks)
Level 5 · 10 marksTry it
2
In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable.

The points $A(-1,\ 4)$ and $B(5,\ -6)$ lie on the straight line $l_1$ .
(a) Find an equation for $l_1$ in the form $ax + by + c = 0$ , where $a$ , $b$ and $c$ are integers.

(3 marks)

The straight line $l_2$ is parallel to the $y$ -axis and passes through $B$ . The point $C$ lies on $l_2$ and the distance from $A$ to $C$ is $\sqrt{85}$
(b) Show that the $y$ coordinate of $C$ satisfies

$y^2 - 8y - 33 = 0$

(3 marks)
(c) Hence find the two possible pairs of coordinates of $C$ .

(2 marks)
(d) Hence find the smaller of the two possible areas of triangle $ABC$ .

(2 marks)
Level 5 · 10 marksTry it