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

Let's suppose that A be (0, 0), B(1, 0) and C(0, 3).
Hence, AB and AC lies on x-axis and y-axis respectively.

Therefore, the equation of circle touching both x-axis and y-axis is as follows :
(x - h)2 + (y - h)2 = h2 (∵ h = k = r)
So, it touches the cirlce as :
$(x-\frac{1}{2})^2+(y-\frac{3}{2})^2=\frac{5}{2}$
Therefore, c1c2 = |r1 - r2|
Now,
$\sqrt{(h-\frac{1}{2})^2+(h-\frac{3}{2})^2}=|h-\frac{\sqrt5}{\sqrt2}|$
$⇒h^2+\frac{1}{4}-h+h^2+\frac{9}{4}-3h$
$=h^2+\frac{5}{2}-\sqrt{10}h$
$⇒h^2+(\sqrt{10}-4)h=0$
$⇒h=4-\sqrt10$
Hence, $r=4-\sqrt{10}=0.84$
∴ the correct answer is 0.84