The equation of the family of curves which intersect the hyperbola $x y=2$ orthogonally is

$y=\frac{x^3}{6}+C$

We have,

$x y=2 \Rightarrow y=\frac{2}{x} \Rightarrow \frac{d y}{d x}=-\frac{2}{x^2}$

Let $y=f(x)$ be the required family of curves. Then,

$\left(\frac{d y}{d x}\right)_{C_1} \times\left(\frac{d y}{d x}\right)_{C_2}=-1$

$\Rightarrow \frac{d y}{d x} \times \frac{-2}{x^2}=-1 \Rightarrow \frac{d y}{d x}=\frac{x^2}{2} \Rightarrow y=\frac{x^3}{6}+C$

This is the required family of curves.