JEE Advanced 2022 Paper 1 · Q08 · Circles

Question

Let $ABC$ be the triangle with $AB=1$, $AC=3$ and $\angle BAC=\dfrac{\pi}{2}$. If a circle of radius $r>0$ touches the sides $AB$, $AC$ and also touches internally the circumcircle of the triangle $ABC$, then the value of $r$ is _____________.

SolveKar AI · Accepted Answer

Place $A$ at origin, $B=(1,0)$, $C=(0,3)$. Circumcircle of right triangle has hypotenuse $BC$ as diameter: centre $M=(1/2,3/2)$, radius $R=\sqrt{10}/2$. A circle touching the two legs $AB$ (x-axis) and $AC$ (y-axis) with centre in the first quadrant has centre $(r,r)$ and radius $r$. For internal tangency with the circumcircle, distance from $(r,r)$ to $M$ equals $R-r$: $\sqrt{(r-1/2)^{2}+(r-3/2)^{2}}=\sqrt{10}/2-r$. Squaring: $2r^{2}-4r+10/4=10/4-r\sqrt{10}+r^{2}\Rightarrow r^{2}+r(\sqrt{10}-4)=0\Rightarrow r=4-\sqrt{10}\approx 0.838$. So $r\approx 0.83$ (key accepts 0.83 or 0.84).

JEE Advanced 2022 Paper 1 · Q08 · Circles

Solution