CAT 2024 Slot 2 QA Q18 Geometry

Question 18 of the Quantitative Ability (QA) section, CAT 2024 Slot 2, with CAT's official answer and a worked solution.

Three circles of equal radii touch (but not cross) each other externally. Two other circles, \(X\) and \(Y\), are drawn such that both touch (but not cross) each of the three previous circles. If the radius of \(X\) is more than that of \(Y\), the ratio of the radii of \(X\) and \(Y\) is

A
\(2+\sqrt{3}: 1\)
B
\(4+\sqrt{3}: 1\)
C
\(7+4 \sqrt{3}: 1\)
D
\(4+2 \sqrt{3}: 1\)

Answer: (C) 7+4 √(3): 1 Geometry MCQ

Solution

Centres of the three equal circles (radius \(r\)) form an equilateral triangle of side \(2r\), with circumradius \(\frac{2r}{\sqrt3}\).
The small circle \(Y\) sits in the middle: radius \(\frac{2r}{\sqrt3} - r\). The large circle \(X\) encloses all three: radius \(\frac{2r}{\sqrt3} + r\).
Ratio \(= \frac{2+\sqrt3}{2-\sqrt3} = (2+\sqrt3)^2 = 7 + 4\sqrt3\).