# Math Help - Question about orthogonal trajectories

1. ## Question about orthogonal trajectories

Find the orthogonal trajectories of the family of ellipses

(x^2/ a^2)+(y^2)=k^2
where a is a given constant.

2. Originally Posted by shannon1111
Find the orthogonal trajectories of the family of ellipses

(x^2/ a^2)+(y^2)=k^2
where a is a given constant.
For orthogonal trajectories, the tangents to the two respective curve are orthogonal. Slope of the tangent for your ellipse is given by

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

For the orthogonal trajectory

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

You now have an ODE you can integrate for the actual curve.

3. If intersted you may want to see the notes and animations on orthogonal trajectories on the

differential equations page of http://calculus7.com/