Example 5. Find the equation of the family of lines orthogonal to the family $x^{2}+y^{2}=2 a x$.
Solution. The given family of lines represents a family of circles, the centers of which lie on the $O x$ axis and which are tangent to the $O y$ axis.

Differentiating both sides of the equation of this family with respect to $x$, we find $x + y y' = a$. By eliminating the parameter $a$ from the equations $x^2 + y^2 = 2ax$ and $x + y y' = a$, we obtain the differential equation of this family: $x^2 - y^2 + 2xy y' = 0$. The differential equation of the orthogonal trajectories is

$$
x^2 - y^2 + 2xy \left(-\frac{1}{y'}\right) = 0, \quad \text{or} \quad y' = \frac{2xy}{x^2 - y^2}
$$

This equation is homogeneous. Integrating it, we find $x^2 + y^2 = Cy$. The integral curves are circles, the centers of which are located on the $O y$ axis and which are tangent to the $O x$ axis (Fig. 16).

![](https://cdn.mathpix.com/cropped/2024_05_22_a0612bb33256580e6dcdg-057.jpg?height=518&width=759&top_left_y=664&top_left_x=307)

Fig. 16