The curve $x = y^2$ and $xy= k$ cut orthogonally, then $k^2$ is equal to:

$\frac{1}{8}$

The correct answer is Option (4) → $\frac{1}{8}$

$\text{Given curves: }x=y^2 \text{ and } xy=k.$

$\text{For }x=y^2:$

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

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

$\text{For }xy=k:$

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

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

$\text{Orthogonal curves satisfy }m_1m_2=-1.$

$\frac{1}{2y}\left(-\frac{y}{x}\right)=-1.$

$-\frac{1}{2x}=-1.$

$x=\frac12.$

$x=y^2 \Rightarrow y^2=\frac12.$

$k=xy=\frac12\cdot y.$

$k^2=x^2y^2=\left(\frac12\right)^2\left(\frac12\right).$

$k^2=\frac18.$

$k^2=\frac18.$