The curves $x=y^2$ and $x y=K$, cut at right angles if:

$8 K^2=1$

The correct answer is Option (2) → $8 K^2=1$

$x=y^2$  ...(1)

$x y=K$   ...(2)

differentiating wrt x

eq. (1)

$1=2y\frac{dy}{dx}$,  $y+x\frac{dy}{dx}=0$

$m_1=\frac{1}{2y}$, $m_2=-\frac{y}{x}$

so $m_1m_2=-1⇒\frac{1}{2x}=1$ so $x=\frac{1}{2}$

from (1) $\frac{1}{2}=y^2$

from (2) $xy=k$

$x^2y^2=k^2⇒(\frac{1}{2})^2×\frac{1}{2}=K^2$

$8K^2=1$