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Equations of Straight Lines — Paper 1 Challenge Questions

Four harder long questions on coordinate geometry of the straight line. Each one states a condition rather than naming a method, and three of the four use a technique the standard eight-question set does not touch at all: a family of lines through a fixed point, reflecting a whole line in another line, and a squared-distance locus that only degenerates into a line under one specific condition. Final answers are on the page. Detailed solutions can be requested by email.

Challenge tier. These questions state a condition rather than naming a method, and several use ideas the standard set does not touch at all. Try the standard set first if you have not already.

4 questions · about 40 minutes, 33 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.

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Question 1 — a family of lines through a fixed point; testing a claim; using the fixed point to answer a follow-up
Question 2 — invariant points under reflection; reflecting a point in a general line; recovering a line from two points
Question 3 — expanding a squared-distance expression symbolically; recognising when a locus is forced to be linear; the algebra of a genuine locus problem
Question 4 — foot of perpendicular from a point to a line; deciding whether a point lies on a segment; the perpendicular-distance formula and when it does not apply

Question 1 · New technique — not tested in the standard set · 10 marks

The equation of the straight line

L

(k + 1) x + (2 - k) y + (k - 8) = 0

, where

k

is a real constant.

(a) Show that $L$ passes through a fixed point $P$ for every value of $k$ , and find the coordinates of $P$ . (4 marks)

(b) Someone claims that $L$ can never be a vertical line, whatever the value of $k$ . Determine whether the claim is correct, explaining your answer. (3 marks)

(c) Find the value of $k$ for which $L$ is parallel to the straight line $3 x + y + 5 = 0$ . (3 marks)

Show answer

(a) $P (2, 3)$

(b) Incorrect. At $k = 2$ , $L$ becomes $3 x - 6 = 0$ , i.e. $x = 2$ , which is vertical.

(c) $k = \frac{5}{4}$

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

Question 2 · New technique — not tested in the standard set · 7 marks

The equation of the straight line

L

2 x - y - 1 = 0

L

is reflected in the straight line

M : x + y - 5 = 0

to give the straight line

L^{'}

(a) Find the coordinates of the point of intersection of $L$ and $M$ . (2 marks)

(b) By reflecting a suitable point on $L$ in $M$ , or otherwise, find the equation of $L^{'}$ . (5 marks)

Show answer

(a) $(2, 3)$

(b) $x - 2 y + 4 = 0$

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

Question 3 · New technique — not tested in the standard set · 7 marks

A (1, 4)

and

B (5, - 2)

are fixed points.

P

is a moving point in the rectangular coordinate plane such that

P A^{2} - m P B^{2} = 20

, where

m

is a positive constant.

(a) Someone claims that the locus of $P$ is a straight line, whatever the value of $m$ . Determine whether the claim is correct, explaining your answer. (3 marks)

(b) In the case where the locus is a straight line, find its equation. (4 marks)

Show answer

(a) Incorrect. Expanding $P A^{2} - m P B^{2}$ gives an $x^{2}$ term and a $y^{2}$ term, each with coefficient $1 - m$ ; these vanish, and the locus is a straight line, only when $m = 1$ .

(b) $2 x - 3 y - 8 = 0$

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

Question 4 · New technique — not tested in the standard set · 9 marks

A (0, 0)

and

B (8, 6)

are fixed points.

C

is the point

(10, - 5)

P

is a point moving along the line segment

A B

, including the two endpoints.

(a) Find the equation of $A B$ . (2 marks)

(b) Find the coordinates of the foot of perpendicular from $C$ to $A B$ , and determine whether this point lies on the segment $A B$ . (4 marks)

(c) Hence, or otherwise, find the least value of $C P$ . (3 marks)

Show answer

(a) $3 x - 4 y = 0$

(b) Foot $= (4, 3)$ ; it is the mid-point of $A B$ , so it lies on the segment.

(c) $10$

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

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

Where this sits in the syllabus

This set sits alongside the standard Paper 1 long questions on the same topic. Where the standard set names a method inside each part, every question here states a condition and leaves the choice of method to you — closer to how recent HKDSE papers, and Paper 1 in particular, have been setting their harder questions.

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

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