Question

A triangle is formed by the tangents at the point $(2,2)$ on the curves $y^2=2 x$ and $x^2+y^2=4 x$, and the line $x+y+2=0$. If $r$ is the radius of its circumcircle, then $r^2$ is equal to

Answer 1

$S_1: y^2=2 x $
$S_2: x^2+y^2=4 x$
$P (2,2)$ is common point on $S _1 \& S _2$
$T_1$ is tangent to $S_1$ at $P $
$\Rightarrow T_1: y \cdot 2=x+2$
$\Rightarrow T_1: x-2 y+2=0$
$T_2$ is tangent to $S_2$ at $P $
$ \Rightarrow T_2: x \cdot 2+y \cdot 2=2(x+2)$
$\Rightarrow T_2: y=2$
$\& L _3: x + y +2=0$ is third line

$ PQ = a =\sqrt{20} $
$ QR = b =\sqrt{8} $
$ RP = c =6$
Area $(\triangle PQR )=\Delta=\frac{1}{2} \times 6 \times 2=6$
$\therefore r =\frac{ abc }{4 \Delta}=\frac{\sqrt{160}}{4}=\sqrt{10} $
$\Rightarrow r ^2=10$