Question

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

• A. $2a$
• B. $a$
• C. $\frac{1}{2}a$
• D. none of these

Answer 1

The correct option is B $a$

Let point $(p,q)$ lies on curve $\sqrt{x}+\sqrt{y}=\sqrt{a}$
therefore $\sqrt{p}+\sqrt{q}=\sqrt{a}$
Now slope $\frac{dy}{dx}=-\sqrt{\frac{p}{q}}$
Equation of tangent is $y-q=-\sqrt{\frac{p}{q}}(x-p)$
X intercept = OP = $\sqrt{p}(\sqrt{p}+\sqrt{q})$
Y intercept = OQ = $\sqrt{q}(\sqrt{p}+\sqrt{q})$
Hence,
OP+OQ $=\sqrt{a}\sqrt{a}=a$