Orthogonal trajectories

Question

Find the orthogonal trajectories to the family of hyperbolas x²-y²=c

Answer 1

Let find differential equation with solution x²-y²=c ;

By differentiation we get: 2x - 2yy'= 0, y' = x/y. Then equation for orthogonal family of lines is y' = -y/x;

we separate variables: dy/y = -dx/x, lny = -lnx + lnC. From here y = C/x.

This is equation of orthogonal family.