Find the shortest chord cut by a line through (1,1).

01 / Standard#Your turn

x^2-4x+y^2=0

Official paper · jm01-2025 · I.10 · PDF 3

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Find the centre and radius.

Hint 2

A shorter chord lies farther from the centre.

Worked solution

Complete squares and compute the centre-to-fixed-point distance.
$(x-2)^2+y^2=4,\quad C=(2,0),\ r=2,\ CP=\sqrt2$
The greatest possible line distance is CP, attained when the line is perpendicular to CP.
$\ell_{\min}=2\sqrt{r^2-CP^2}=2\sqrt2$

E: 2√2.

Checks and common pitfalls: The fixed point lies inside the circle, so a tangent through it is impossible.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
Complete squares and compute the centre-to-fixed-point distance.
$(x-2)^2+y^2=4,\quad C=(2,0),\ r=2,\ CP=\sqrt2$
The greatest possible line distance is CP, attained when the line is perpendicular to CP.
$\ell_{\min}=2\sqrt{r^2-CP^2}=2\sqrt2$

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

2025 JM01