Consider a circle (x - α ^ 2 ) + (y - β ^ 2 ) = 50 , where α, β > 0...

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.

$⇒ CP ⊥ \left(x + y = 0\right)$ and $CP = r$

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

$⇒ 5\sqrt{2} = \left|\dfrac{\alpha + \beta}{\sqrt{2}}\right|$

$⇒ 50 = \dfrac{{\left(\alpha + \beta\right)}^{2}}{2}$

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