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GATE 2026 XH5 (Psychology) – Question 10

General Aptitude · Quantitative Aptitude: Algebra, Geometry and Mensuration · 2 marks · Multiple choice

As shown in the figure, circle $C_1$ with center $O_1$ and radius $r_1$ touches the square $VWXY$ at points $P$ and $Q$ while circle $C_2$ with center $O_2$ and radius $r_2$ touches the square $VWXY$ at points $R$ and $S$. The two circles touch each other at $T$.

Given $r_1 = 1$ cm and $\overline{VY} = \overline{VW} = 4$ cm, $r_2 =$ _____ cm.

A square VWXY of side 4 cm. A small circle of radius 1 cm sits in the lower left corner, touching the left side at P and the bottom side at Q. A larger circle sits in the upper right corner, touching the top side at S and the right side at R. The two circles touch at T.
  1. $4 - 3\sqrt{2}$
  2. $1 + 2\sqrt{2}$
  3. $7 - 4\sqrt{2}$
  4. $5 + 3\sqrt{2}$

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Correct answer: (C) $7 - 4\sqrt{2}$

Explanation

Take the lower left corner as the origin. $C_1$ touches the left and bottom sides, so $O_1 = (1, 1)$. $C_2$ touches the top and right sides, so $O_2 = (4 - r_2, 4 - r_2)$. The circles touch externally, so $O_1O_2 = 1 + r_2$. Now $O_1O_2 = \sqrt{2}\,(3 - r_2)$, so $\sqrt{2}(3 - r_2) = 1 + r_2$, which gives $r_2 = \frac{3\sqrt{2} - 1}{\sqrt{2} + 1} = (3\sqrt{2} - 1)(\sqrt{2} - 1) = 7 - 4\sqrt{2}$ cm (about 1.34 cm).