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 ______.
Step-by-Step Solution
Key Concept: Find the tangent to $y^2 = 2x$ at $(2,2)$: $T_1: x - 2y + 2 = 0$. Tangent to $x^2+y^2=4x$ at $(2,2)$: $T_2: y = 2$. Third line $L_3: x+y+2=0$. Find vertices, compute sides and area, then $r = \frac{abc}{4\Delta}$.
Three lines: $x-2y+2=0$, $y=2$, $x+y+2=0$. Vertices: $P(2,2)$, $Q(-2,0)$, $R(-4,2)$. $r = \sqrt{10}$, $r^2 = 10$.
Correct Answer: 10