Given: $\left(x-{\alpha }^2\right)+\left(y-{\beta }^2\right)=50$

The given equation of circle represent centre as $\left(\alpha ,\beta \right)$ and radius as $5\sqrt{2}$ units.

Now, $x+y=0$ is tangent to the given circle at $P$.

We know that, radius is perpendicular to tangent at the point of tangency.

$\Rightarrow CP\perp \left(x+y=0\right)$ and $CP=r$

$\Rightarrow r=\left|\frac{\alpha \times 1+\beta \times 1}{\sqrt{1^2+1^2}}\right|$

$\Rightarrow 5\sqrt{2}=\left|\frac{\alpha +\beta }{\sqrt{2}}\right|$

$\Rightarrow 50=\frac{{\left(\alpha +\beta \right)}^2}{2}$

$\Rightarrow {\left(\alpha +\beta \right)}^2=100$