Question

Find the orthogonal trajectories of the circles x² + y² – ay = 0, where a is a parameter.

Answer 1

Given curve is

x² + y² – ay = 0

Eliminating a between (i) and (ii), we get

Replace dy/dx by – dx/dy, we get

Integrating, we get

log |v² + 1| + log |x| = log c

x(v² + 1) = c

x² + y² – cx = 0

which is the required orthogonal trajectories.