Orthogonal trajectories . eliminating parameter

Question

Find the orthogonal trajectories of $${x^{2} \over a^{2}} + {y^{2} \over a^{2}+t}= 1\,,\qquad t\quad \mbox{is a parameter} $$

I've tried eliminating $t$ and its given me a complicated looking equation. Surely I'm doing something wrong.

Answer 1

I am giving a try here. Implicit diff gives: $y' = - \frac{x(a^2+t)}{ya^2}$ For orthogonal trajectories, take opposite reciprocal to get $y' = \frac{ya^2}{x(a^2+t)}$ By seperation of variables you get $\frac{y'}{ya^2}=\frac{1}{x(a^2+t)}$ Anti deriving both sides gives $\frac{ln|y|}{a^2} = \frac{ln|x|}{a^2+t}+C$ From which it becomes algebra to solve for y if you wish. Hope this is what you are looking for.