If the curves $x=y^2$ and $xy=k$ meet at right angles, then the value of k is :

$±\frac{1}{2\sqrt{2}}$

The correct answer is Option (4) → $±\frac{1}{2\sqrt{2}}$

$x=y^2$  ...(1)

$xy=k$   ...(2)

$2y\frac{dy}{dx}=1$

$⇒\frac{dy}{dx}=\frac{1}{2y}$

$y+x\frac{dy}{dx}=0$

$\frac{dy}{dx}=-\frac{y}{x}$

$⇒\frac{1}{2y}×-\frac{y}{x}=-1$

so $x=\frac{1}{2}$

from (1) $y=±\frac{1}{\sqrt{2}}$

from (2) $k=±\frac{1}{2\sqrt{2}}$