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Equations of Straight Lines — Paper 2 Challenge MC

Eight harder multiple-choice questions on coordinate geometry of the straight line, built around ideas the standard twenty-question set leaves out: a line that passes through a fixed point whatever a parameter is, a squared-distance locus that only becomes a line under one condition, and a minimum-distance question with a genuine trap built in. Answers are explained on the page.

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.

8 questions · about 20 minutes · 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
Question 2 — recognising when a squared-distance locus is forced to be linear
Question 3 — reflecting a point in a general (non-axis, non-45°) line
Question 4 — checking whether the foot of a perpendicular lies on a segment before trusting the perpendicular-distance formula
Question 5 — solving a parallel condition and then rejecting the root that makes the lines identical
Question 6 — finding an intersection point first, then using it as a condition on a third line
Question 7 — recognising when a locus is a two-dimensional region rather than a curve
Question 8 — a family of lines through a fixed point, combined with reasoning about which member of the family meets a given segment

Question 1 · New technique

The equation of the straight line

L

(2 k - 1) x + (k + 3) y + (6 - 5 k) = 0

, where

k

is a real constant.

L

passes through a fixed point

P

for every value of

k

. The coordinates of

P

are

$(3, - 1)$ .
$(- 3, 1)$ .
$(3, 1)$ .
$(- 3, - 1)$ .

Show answer

A. Group the equation by $k$ : $(2 x + y + 6) + k (- x + y - 5) = 0$ . For this to hold for every $k$ , both brackets must vanish: $2 x + y + 6 = 0$ and $- x + y - 5 = 0$ , giving $(3, - 1)$ .

Question 2 · New technique

A

and

B

are fixed points, and

P

is a moving point such that

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

, where

m

is a positive constant and

c

is a constant. For which value of

m

is the locus of

P

always a straight line, whatever the value of

c

$1$
$2$
$\frac{1}{2}$
It cannot be determined without knowing $A$ and $B$ .

Show answer

A. Expanding $P A^{2} - m P B^{2}$ always produces an $x^{2}$ term and a $y^{2}$ term, each with coefficient $1 - m$ . These vanish, leaving a linear equation, exactly when $m = 1$ — for any other $m$ the locus is generally a circle.

Question 3 · New technique

The image of the point

(1, 2)

under reflection in the straight line

x - y - 3 = 0

$(2, 1)$ .
$(5, - 2)$ .
$(- 2, 5)$ .
$(4, - 1)$ .

Show answer

B. The segment joining $(1, 2)$ to its image is perpendicular to $x - y - 3 = 0$ and its mid-point lies on that line. Solving these two conditions gives the image $(5, - 2)$ . Option A only swaps the coordinates without accounting for the line's position.

Question 4 · New technique

P

is a moving point on the line segment joining

O (0, 0)

and

B (8, 6)

C

is the point

(- 1, - 7)

. The least value of

C P

$5$ .
$510$ .
$7$ .
$52$ .

Show answer

D. The foot of the perpendicular from $C$ to line $O B$ is $(- 4, - 3)$ , which lies beyond $O$ — outside the segment. The nearest point on the segment is therefore the endpoint $O$ , and the least value of $C P$ is $C O = 52$ , not the perpendicular distance. Option A, $5$ , is exactly that perpendicular distance — the right number for the wrong question.

Question 5 · New technique

The equations of the straight lines

L_{1}

and

L_{2}

are

x + m y + 1 = 0

and

m x + 4 y + 2 = 0

respectively, where

m

is a constant. If

L_{1}

and

L_{2}

are parallel and distinct, then

m =

$- 2$ .
$2$ .
$- 2$ or $2$ .
$4$ .

Show answer

A. Parallel requires $1 \times 4 = m \times m$ , so $m = \pm 2$ . At $m = 2$ , $L_{1}$ and $L_{2}$ become the same line, $x + 2 y + 1 = 0$ scaled — not distinct, so it is rejected. Only $m = - 2$ gives lines that are parallel and different.

Question 6 · New technique

The straight lines

x - 2 y + 3 = 0

and

2 x + y - 4 = 0

intersect at the point

P

. If the straight line

3 x + k y - 7 = 0

also passes through

P

, then

k =

$- 2$
$- 1$
$1$
$2$

Show answer

D. Solving the first two equations gives $P (1, 2)$ . Substituting into the third line: $3 (1) + 2 k - 7 = 0$ , so $k = 2$ .

Question 7 · New technique

The equations of the straight lines

L_{1}

and

L_{2}

are

3 x + 4 y - 12 = 0

and

3 x + 4 y + 8 = 0

respectively.

P

is a moving point such that the sum of the perpendicular distances from

P

L_{1}

and to

L_{2}

is always

4

. The locus of

P

a single point.
a straight line parallel to $L_{1}$ and $L_{2}$ .
a pair of straight lines.
the region between $L_{1}$ and $L_{2}$ , including both lines.

Show answer

D. $L_{1}$ and $L_{2}$ are parallel, and the perpendicular distance between them is exactly $4$ . For any point lying between the two lines, the sum of its distances to both is always this same constant $4$ ; a point outside the strip gives a sum greater than $4$ . The condition therefore describes the whole strip between the lines, not a single curve.

Question 8 · New technique

A (- 2, - 1)

and

B (4, 5)

are fixed points. For how many integer values of

k

does the straight line

y = k x + 3

pass through a point on the line segment

A B

, including the two endpoints?

Show answer

B. Every such line passes through $(0, 3)$ , a fixed point not on $A B$ . As the line turns from the one through $A$ (slope $2$ ) to the one through $B$ (slope $\frac{1}{2}$ ), it sweeps across every point of the segment, so it meets $A B$ exactly when $\frac{1}{2} \leq k \leq 2$ . The integers in this range are $k = 1$ and $k = 2$ , so there are $2$ values.

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

Where this sits in the syllabus

This set sits alongside the standard Paper 2 multiple-choice questions on the same topic. Two of these — the segment-distance question and the parallel-and-distinct question — are built specifically to reward checking your answer against a second condition rather than stopping at the first formula that fits.

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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