Find the orthogonal trajectories of the family of ellipses

(x^2/ a^2)+(y^2)=k^2

where a is a given constant.

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- Jul 14th 2009, 01:34 PMshannon1111Question 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. - Jul 14th 2009, 01:58 PMJester
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. - Jul 16th 2009, 09:58 AMCalculus26
If intersted you may want to see the notes and animations on orthogonal trajectories on the

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