For M on the x-axis, minimise PM+QM using P=(0,5), Q=(2,9), and find M.

01 / Standard#Your turn

Official paper · jm01-2026 · II.2(b) · PDF 4

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Reflect one endpoint across the x-axis.

Hint 2

Use the straight segment from the reflected point to the other endpoint.

Worked solution

Reflect P to P′; distances to points on the x-axis are unchanged.
$P'=(0,-5),\quad PM=P'M,\quad P'M+MQ\ge P'Q$
Equality holds where the segment crosses the axis.
$P'Q:\ y=7x-5,\quad M=\left(\frac57,0\right)$
Compute the straight-line distance.
$\min(PM+QM)=\sqrt{2^2+14^2}=10\sqrt2$

Minimum 10√2, attained at M=(5/7,0).

Checks and common pitfalls: Connecting P directly to Q does not cross the x-axis and cannot realise the constrained path.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
Reflect P to P′; distances to points on the x-axis are unchanged.
$P'=(0,-5),\quad PM=P'M,\quad P'M+MQ\ge P'Q$
Equality holds where the segment crosses the axis.
$P'Q:\ y=7x-5,\quad M=\left(\frac57,0\right)$
Compute the straight-line distance.
$\min(PM+QM)=\sqrt{2^2+14^2}=10\sqrt2$

Think first. Reveal a hint when the class is ready.

2026 JM01