Question

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

Answer 1

Answer: 10

$S_1:y^2=2x$
$S_2:x^2+y^2=4x$
$P(2,2)$ is common point on $S_1\&S_2$
$T_1$ is tangent to $S_1$ at $P$
$\Rightarrow T_1:y\cdot 2=x+2$
$\Rightarrow T_1:x-2y+2=0$
$T_2$ 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 $(△PQR)=\Delta =\frac{1}{2}\times 6\times 2=6$
$\therefore r=\frac{abc}{4\Delta }=\frac{\sqrt{160}}{4}=\sqrt{10}$
$\Rightarrow r^2=10$