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
$\displaystyle
\frac{dy}{dx} = - \frac{x}{a^2 y}
$
For the orthogonal trajectory
$\displaystyle
\frac{dy}{dx} = \frac{a^2 y}{x}
$
You now have an ODE you can integrate for the actual curve.
If intersted you may want to see the notes and animations on orthogonal trajectories on the
differential equations page of http://calculus7.com/