Question

If the tangent to the curve $\sqrt{x}+\sqrt{y}=\sqrt{a}$ at any point on it cuts the axes OX and OY at P and Q respectively, then OP +OQ is

• A. a/2
• B. a
• C. 2a
• D. 4a

Answer 1

The correct option is B

a

$\sqrt{x}+\sqrt{y}=\sqrt{a}.....\left(i\right)$
$\Rightarrow \frac{1}{2\sqrt{x}}+\frac{1}{2\sqrt{y}}\frac{dy}{dx}=0$

$\therefore \frac{dy}{dx}=-\frac{\sqrt{y}}{\sqrt{x}}$
Equation of tangent at $\left(x_1{y}_1\right)isy-y_1=-\frac{\sqrt{y_1}}{\sqrt{x_1}}(x-x_1)$
$\Rightarrow \frac{x}{\sqrt{x_1}}+\frac{y}{\sqrt{y_1}}=\sqrt{a};\Rightarrow op=\sqrt{a}\sqrt{x_1},OQ=\sqrt{a}\sqrt{y_1}\therefore OP+OQ=\sqrt{a}$