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