Find the minimum of |z−z₁| on the locus.

01 / Standard#Your turn

z_1=3+5i,\quad z_2=5+i,\quad|z-z_1|=|z-z_2|

Official paper · jm02-2022 · 4(a)(iii) · PDF 6

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

The nearest point of the perpendicular bisector is the segment midpoint.

Hint 2

The minimum distance is half |z₂−z₁|.

Worked solution

Compute the distance between the fixed points.
$|z_2-z_1|=|2-4i|=\sqrt{20}=2\sqrt5$
Halve it, attained at z=4+3i.
$\min|z-z_1|=\sqrt5$

Minimum √5 at z=4+3i.

Checks and common pitfalls: The endpoint z₁ itself is not equidistant from the two fixed points.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
Compute the distance between the fixed points.
$|z_2-z_1|=|2-4i|=\sqrt{20}=2\sqrt5$
Halve it, attained at z=4+3i.
$\min|z-z_1|=\sqrt5$

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

2022 JM02