A triangle is formed by the tangents at the point (2,2) on the curves ${\mathrm{y}}^2=2\mathrm{x}$ and ${\mathrm{x}}^2+{\mathrm{y}}^2=4\mathrm{x}$   and the line $\mathrm{x}+\mathrm{y}+2=0$. If r is the radius of its circumcircle, then r²  is equal to _____.

$$\begin{gathered} \begin{array}{l}Equation of \tan gent at \left(2 ,2\right) \\ is 2y=x+2\text{ for }{y}^2=2x \\ Equation of \tan gent at \left(2 ,2\right)\\ is 2x+2y=2(x+2)\text{ for }{x}^2+y^2=4x\\ \Rightarrow y=2, \\ Given line is x+y+2=0\end{array} \end{gathered}$$

To find circumcenter Find the perpendicular bisectors of AB & AC
For AB it is x=-1
For BC
$\begin{array}{l}(\mathrm{x}+4{)}^2+(\mathrm{y}-2{)}^2=(\mathrm{x}+2{)}^2+{\mathrm{y}}^2\\ 8\mathrm{x}-4\mathrm{y}+20=4\mathrm{x}+4\Rightarrow 4\mathrm{x}-4\mathrm{y}+16=0\Rightarrow \mathrm{x}-\mathrm{y}+4=0\end{array}$
$\therefore \text{ Centre }(-1,3)\text{ Radius }{r}^2=10$