Practice › Measures, Shape and Space › Equations of Straight Lines

Equations of Straight Lines — Paper 1 Long Questions

Eight original long questions on coordinate geometry of the straight line, written to match the shape and difficulty of HKDSE Paper 1. They run from a four-mark warm-up on the two-point form up to an eleven-mark question where the answer stays in terms of m. Final answers are on the page. Detailed solutions can be requested by email.

8 questions · about 75 minutes, 61 marks · answers below each question

All questions are original, written for Math and AI Academy and modelled on past-paper style. They are not reproductions of HKEAA examination questions.

Jump to

Question 1 — two-point form; parallel lines share a slope
Question 2 — intercepts of a line in general form; area of a right-angled triangle; solving for a constant
Question 3 — reading intercepts off a figure; perpendicular slopes; area from intercepts
Question 4 — simultaneous equations; concurrency; a line with equal intercepts
Question 5 — testing parallelism; locus of a point equidistant from two parallel lines; intercepts and area
Question 6 — locus of a point equidistant from two fixed points; intersection of two lines; area of a triangle from coordinates
Question 7 — perpendicular bisectors; circumcentre as their intersection; using side lengths to decide whether a triangle is acute
Question 8 — intersection of a line through the origin with a fixed line; areas in terms of a parameter; triangles with a common vertex and bases on one line

Question 1 · Modelled on 2026 P1 Q12(a) · 4 marks

The straight line

L_{1}

passes through the points

A (- 3, 8)

and

B (5, - 4)

(a) Find the equation of $L_{1}$ . (2 marks)

(b) The straight line $L_{2}$ passes through the origin and is parallel to $L_{1}$ . Find the equation of $L_{2}$ . (2 marks)

Show answer

(a) $3 x + 2 y - 7 = 0$

(b) $3 x + 2 y = 0$

Final answers only. Detailed solutions can be requested by email.

Question 2 · Modelled on 2023 P2 Q10 and 2018 P2 Q26 · 5 marks

The straight line

L : 2 x - 5 y + k = 0

, where

k

is a positive constant, cuts the

x

-axis at

A

and cuts the

y

-axis at

B

. Denote the origin by

O

. It is given that the area of

△ O A B

45

(a) Express the coordinates of $A$ and of $B$ in terms of $k$ . (2 marks)

(b) Find $k$ . (3 marks)

Show answer

(a) $A (- \frac{k}{2}, 0)$ , $B (0, \frac{k}{5})$

(b) $k = 30$

Final answers only. Detailed solutions can be requested by email.

Question 3 · Modelled on 2019 P2 Q23 · 7 marks

In the figure, the straight line

L_{1} : a x + b y + 12 = 0

cuts the

x

-axis at

(- 4, 0)

and cuts the

y

-axis at

(0, 6)

, where

a

and

b

are constants.

(a) Find $a$ and $b$ . (3 marks)

(b) The straight line $L_{2}$ is perpendicular to $L_{1}$ and passes through $(0, 6)$ . $L_{2}$ cuts the $x$ -axis at $C$ . Find the coordinates of $C$ . (2 marks)

(c) Find the area of the triangle bounded by $L_{1}$ , $L_{2}$ and the $x$ -axis. (2 marks)

Show answer

(a) $a = 3$ , $b = - 2$

(b) $C (9, 0)$

(c) $39$

Final answers only. Detailed solutions can be requested by email.

Question 4 · Modelled on 2022 P2 Q18 · 7 marks

The equations of the straight lines

L_{1}

L_{2}

and

L_{3}

are

x + 2 y - 8 = 0

3 x - y - 3 = 0

and

k x + y - 11 = 0

respectively, where

k

is a constant.

(a) Find the coordinates of the point of intersection $P$ of $L_{1}$ and $L_{2}$ . (3 marks)

(b) It is given that $L_{1}$ , $L_{2}$ and $L_{3}$ are concurrent. Find $k$ . (2 marks)

(c) Find the equation of the straight line which passes through $P$ and whose $x$ -intercept and $y$ -intercept are equal and non-zero. (2 marks)

Show answer

(a) $P (2, 3)$

(b) $k = 4$

(c) $x + y - 5 = 0$

Final answers only. Detailed solutions can be requested by email.

Question 5 · Modelled on 2026 P2 Q26 and 2020 P1 Q14(b) · 8 marks

The equations of the straight lines

L_{1}

and

L_{2}

are

3 x - 4 y + 12 = 0

and

3 x - 4 y - 36 = 0

respectively.

(a) Someone claims that $L_{1}$ and $L_{2}$ are parallel. Do you agree? Explain your answer. (2 marks)

(b) $P$ is a moving point in the rectangular coordinate plane such that the perpendicular distance from $P$ to $L_{1}$ is equal to the perpendicular distance from $P$ to $L_{2}$ . Describe the locus of $P$ and find its equation. (3 marks)

(c) The locus of $P$ cuts the $x$ -axis at $A$ and cuts the $y$ -axis at $B$ . Find $A B$ and the area of $△ O A B$ , where $O$ is the origin. (3 marks)

Show answer

(a) Agree. Both slopes are $\frac{3}{4}$ and the $y$ -intercepts differ.

(b) The locus is the straight line midway between $L_{1}$ and $L_{2}$ : $3 x - 4 y - 12 = 0$

(c) $A B = 5$ ; area $= 6$

Final answers only. Detailed solutions can be requested by email.

Question 6 · Modelled on 2015 P1 Q14(a) and 2023 P2 Q26 · 9 marks

The coordinates of the points

A

and

B

are

(- 6, 3)

and

(4, - 7)

respectively.

P

is a moving point in the rectangular coordinate plane such that

A P = B P

. Denote the locus of

P

Γ

(a) Find the equation of $Γ$ . (3 marks)

(b) $Γ$ cuts the straight line $x + 2 y - 7 = 0$ at $Q$ . Find the coordinates of $Q$ . (3 marks)

(c) Find the area of $△ A B Q$ . (3 marks)

Show answer

(a) $x - y - 1 = 0$

(b) $Q (3, 2)$

(c) $40$

Final answers only. Detailed solutions can be requested by email.

Question 7 · Modelled on 2016 P1 Q20 and 2023 P1 Q19 · 10 marks

In the figure, the coordinates of the points

A

B

and

C

are

(- 2, 6)

(7, 3)

and

(2, - 2)

respectively.

(a) Find the equation of the perpendicular bisector of $A B$ . (3 marks)

(b) Find the equation of the perpendicular bisector of $B C$ . (2 marks)

(c) Hence find the coordinates of the circumcentre $D$ of $△ A B C$ . (3 marks)

(d) Is $D$ an interior point of $△ A B C$ ? Explain your answer. (2 marks)

Show answer

(a) $3 x - y - 3 = 0$

(b) $x + y - 5 = 0$

(c) $D (2, 3)$

(d) Yes. $A B^{2} = 90$ , $B C^{2} = 50$ , $C A^{2} = 80$ ; every angle is acute, so the circumcentre lies inside.

Final answers only. Detailed solutions can be requested by email.

Question 8 · Modelled on 2024 P1 Q12(a) · 11 marks

In the figure, the straight line

L_{1} : y = m x

, where

m > 0

, and the straight line

L_{2} : x + y - 6 = 0

intersect at

P

L_{2}

cuts the

x

-axis at

A

and cuts the

y

-axis at

B

. Denote the origin by

O

(a) Write down the coordinates of $A$ and of $B$ . (2 marks)

(b) Express the coordinates of $P$ in terms of $m$ . (3 marks)

(c) Express the area of $△ O A P$ and the area of $△ O B P$ in terms of $m$ . (3 marks)

(d) It is given that the area of $△ O A P :$ the area of $△ O B P = 2 : 1$ . Find $m$ . Hence write down $A P : P B$ . (3 marks)

Show answer

(a) $A (6, 0)$ , $B (0, 6)$

(b) $P (\frac{6}{m + 1}, \frac{6 m}{m + 1})$

(c) Area of $△ O A P = \frac{18 m}{m + 1}$ ; area of $△ O B P = \frac{18}{m + 1}$

(d) $m = 2$ ; $A P : P B = 2 : 1$

Final answers only. Detailed solutions can be requested by email.

Continue practising this topic: Try the Paper 1 Challenge for harder variants, or move to Paper 2 multiple-choice — also available as Paper 2 Challenge MC.

Where this sits in the syllabus

Equations of straight lines sits in the Measures, Shape and Space strand. It is the language every later coordinate topic is written in: equations of circles, loci and linear programming all assume you can move between a slope, two points and a general-form equation without thinking about it. Most marks lost here are lost on signs, not on method.

Full coverage of this topic, and the rest of the course, is on the Mathematics page. More sets are listed on the practice index.

Start with a free discovery session

No cost, no obligation. We look at where your child actually is, and show you what a term of real progress would look like.

Book a free discovery session See the programmes