Question

Let $A B C$ be the triangle with $A B=1, A C=3$ and $\angle B A C=\frac{\pi}{2}$. If a circle of radius $r>0$ touches the sides $A B, A C$ and also touches internally the circumcircle of the triangle $A B C$, then the value of $r$ ___

Answer 1

$C _1\left(\frac{1}{2}, \frac{3}{2}\right)$ and $r _1=\frac{\sqrt{10}}{2}$
$C _2=( r , r )$
$\therefore$ circle $C _2$ touches $C _1$ internally
$ \Rightarrow C _1 C _2=\left| r -\frac{\sqrt{10}}{2}\right| $
$ \Rightarrow\left( r -\frac{1}{2}\right)^2+\left( r -\frac{3}{2}\right)^2=\left( r -\frac{\sqrt{10}}{2}\right)^2 $
$r ^2-4 r +\sqrt{10} r =0 $
$ r =0 \text { (reject) or } r =4-\sqrt{10}$