Find the orthogonal trajectories of the family of curves consisting of all circles passing through the points 
and 
.
In a previous exercise (section 8.22, Exercise #11) we found that the family of all circles passing through the points and satisfy the differential equation
![\[ (x^2 - y^2 -1) y' - 2xy = 0 \quad \implies \quad y' = \frac{2xy}{x^2-y^2-1}. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-6d72bf72d63a55e89723929312655089_l3.png "Rendered by QuickLaTeX.com")
Therefore, the orthogonal trajectories satisfy the differential equation
![\[ y' = \frac{2xy}{y^2 - x^2 + 1}. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-a6121290e340be6f6f6446737de5b638_l3.png "Rendered by QuickLaTeX.com")
Incomplete.
It is easy to solve with substitution: $v=y^2$.