Two externally tangent circles of radii 3 and 4 touch the same straight line. A smaller circle occupies the bounded gap and is tangent to both and to the line. Find its radius.

01 / Standard#Your turn

Official paper · jm01-2022 · I.7 · PDF 2

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

For radii R,r tangent to the line, the horizontal centre separation is 2√(Rr).

Hint 2

The small-circle centre lies horizontally between the large centres.

Worked solution

Derive and add the two horizontal gaps.
$d^2=(R+r)^2-(R-r)^2=4Rr,\quad2\sqrt{3r}+2\sqrt{4r}=2\sqrt{12}$
Solve for r and rationalise.
$r=\frac{12}{(\sqrt3+2)^2}=\frac{12}{7+4\sqrt3}=84-48\sqrt3$

A: 84−48√3.

Checks and common pitfalls: The bounded-gap condition selects the sum of the two horizontal separations.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
Derive and add the two horizontal gaps.
$d^2=(R+r)^2-(R-r)^2=4Rr,\quad2\sqrt{3r}+2\sqrt{4r}=2\sqrt{12}$
Solve for r and rationalise.
$r=\frac{12}{(\sqrt3+2)^2}=\frac{12}{7+4\sqrt3}=84-48\sqrt3$

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

2022 JM01