GATE 2026 IN – Question 10
Circles $C_1$, $C_2$, and $C_3$, with centers $O_1$, $O_2$, and $O_3$, and radii $r_1$, $r_2$, and $r_3$, respectively, touch each other as shown in the figure. Given $r_1=2\ \mathrm{cm}$, $r_2=1\ \mathrm{cm}$, and $\angle O_1O_3O_2=90^\circ$, $r_3=$ ________ cm.
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Correct answer: (A) $\frac{-3+\sqrt{17}}{2}$
Explanation
For externally tangent circles, the distance between their centers equals the sum of their radii. Writing $r=r_3$ in cm gives $O_1O_2=3$, $O_1O_3=2+r$, and $O_2O_3=1+r$.
The angle at $O_3$ is a right angle, so the Pythagorean theorem gives $(2+r)^2+(1+r)^2=3^2$. Expanding and simplifying gives $2r^2+6r-4=0$, or $r^2+3r-2=0$.
The roots are $r=(-3\pm\sqrt{17})/2$. A radius must be positive, so $r_3=(-3+\sqrt{17})/2\ \mathrm{cm}$. The correct option is **A**.