Question

Find the orthogonal trajectories of the family of the curves ax² + y² = 1.

Answer 1

The given curve is

ax² + y² = 1 ...(i)

2ax + 2y(dy/dx) = 0

ax + y(dy/dx) = 0 ...(ii)

Solving Eqs (i) and (ii), we get 

Integrating, we get

(x²/2) – (y²/2) + log |y| = c

(x² – y²) + 2log |y| = c

which is the required orthogonal trajectories.