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GATE 2026 CH – 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 with V at top left and X at bottom right. A small circle of radius 1 cm sits in the bottom left corner, touching sides VW and WX. A larger circle sits in the top right corner touching sides VY and YX, and the two circles touch each other.
  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

Put W at the origin. The centre of $C_1$ is $(1, 1)$ and the centre of $C_2$ is $(4 - r_2, 4 - r_2)$. The circles touch, so the distance between the centres is $1 + r_2$: $\sqrt{2}(3 - r_2) = 1 + r_2$. Solving, $r_2 = \frac{3\sqrt{2} - 1}{\sqrt{2} + 1} = (3\sqrt{2} - 1)(\sqrt{2} - 1) = 7 - 4\sqrt{2} \approx 1.34$ cm.