Sum of the intercepts on the co-ordinate axes of any tangent to $\sqrt{x}+\sqrt{y}=\sqrt{a}$  is:

$\sqrt{x}+\sqrt{y}=\sqrt{a}\Rightarrow \frac{dy}{dx}=-\frac{\sqrt{y}}{\sqrt{x}}$

$\therefore$     Equation of tangent at $\left(x_1,y_1\right)$ is$y-y_1=-\frac{\sqrt{y_1}}{\sqrt{x_1}}\left(x-x_1\right)$

$\alpha =$ intercept on $x-$ axis $=\sqrt{a}\sqrt{x_1}$

$\beta =$ intercept on $y-$ axis $=\sqrt{a}\sqrt{y_1}$

$\therefore$    $\alpha +\beta =\sqrt{a}(\sqrt{x_1}+\sqrt{y_1})=\sqrt{a}\sqrt{a}=a$ .