The sum of intercepts on coordinate axes made by tangent to the curve \(\sqrt{x}+\sqrt{y}=\sqrt{a}\) is

\(\sqrt{x}+\sqrt{y}=\sqrt{a}\)
Differentiating w.r.t. \(x\), we get
\(\frac{1}{2 \sqrt{x}}+\frac{1}{2 \sqrt{y}} \frac{\mathrm{~d} y}{\mathrm{~d} x}=0\)
\(\therefore \) Slope of the tangent is \(\frac{\mathrm{d} y}{\mathrm{~d} x}=-\sqrt{\frac{y}{x}}\)
\(\therefore \) Equation of the tangent at \(\left(x_1, y_1\right)\) is
\(\left(y-y_1\right)=-\sqrt{\frac{y_1}{x_1}}\left(x-x_1\right)\)
\(\sqrt{x_1} y+\sqrt{y_1} x=\sqrt{x_1} y_1+\sqrt{y_1} x_1 \)
\( \therefore \sqrt{y_1} x+\sqrt{x_1} y=\sqrt{x_1 y_1}\left(\sqrt{x_1}+\sqrt{y_1}\right) \)
\( \therefore \sqrt{y_1} x+\sqrt{x_1} y=\sqrt{x_1 y_1 \mathrm{a}} \)
\( \therefore \frac{x}{\sqrt{x_1 \mathrm{a}}}+\frac{y}{\sqrt{y_1 \mathrm{a}}}=1 \)
\( \therefore x \text {-intercept }+y \text {-intercept }=\sqrt{\mathrm{a}}\left(\sqrt{x_1}+\sqrt{y_1}\right) \)
\( =\sqrt{\mathrm{a}} \times \sqrt{\mathrm{a}}=\mathrm{a}\)