Answer
$$
x^{2}-2y^{2}=C, \quad\quad \text{(hyperbolas)}
$$
the orthogonal trajectories of the given family are
$$
y=x^{-2}. K
$$
where $K$ is an arbitrary constant.
Work Step by Step
$$
x^{2}-2y^{2}=C, \quad\quad \text{(hyperbolas)}
$$
First, solve the given equation by differentiating implicitly with respect to $x$ we obtain the differential equation:
$$
2 x -4y y^{\prime}=0
$$
Slope of given family is
$$
y^{\prime}=\frac{x}{2y}
$$
Because represents the slope of the given family of curves at it follows that
the orthogonal family has the negative reciprocal slope So,
$$
y^{\prime}=\frac{-2y}{x}
$$
Now you can find the orthogonal family by separating variables and integrating.
$$
\int \frac{dy}{y}=\int\frac{-2dx}{x}
$$
$\Rightarrow$
$$
\ln |y|=-2\ln | x|+\ln K \Rightarrow \ln |y|=\ln | x^{-2}K|
$$
$\Rightarrow$
$$
y=x^{-2}. K
$$
where $K$ is an arbitrary constant.
