# Math Help - Orthogonal Trajectories

1. ## Orthogonal Trajectories

Hi, how do I find the orthogonal trajectories of the family of ellipses:

$\frac{x^2}{a^2} + y^2 = k^2$

where a is a constant?

Thanks.

2. Originally Posted by coldfire
Hi, how do I find the orthogonal trajectories of the family of ellipses:

$\frac{x^2}{a^2} + y^2 = k^2$

where a is a constant?

Thanks.
First calculate the slope of the tangent

$
\frac{2x}{a^2} + 2 y \frac{dy}{dx} = 0 \;\text{or}\; \frac{dy}{dx} = - \frac{x}{a^2 y}.
$

Orthogonal trajectories will be perpendicular so

$\frac{dy}{dx} = \frac{a^2y}{x}$.

Now integrate.