Orthogonal Trajectories

Some of the curves in this family are shown at right.

We now wish to find another family or curves that will be orthogonal to each of these ellipses wherever they intersect. This new family is known as the orthogonal trajectories of the ellipses.

First we differentiate the given family of curves implicitly to get

Our new family must solve the differential equation

This is separable.

This is a power function. These curves will intersect the ellipses everywhere at right angles. You can see the graphs of both families at right.

STEP 2001, Math III, #7: Sketch the graph of the function

.

Show that the differential equation

describes a family of parabolas each of which passes through the points and and has its vertex on the -axis.

Hence find the equation of the curve that passes through the point and intersects each of the above parabolas orthogonally. Sketch this curve.

[Two curves intersect orthogonally if their tangents at the point of intersection are perpendicular.]

Alternative Coordinate Systems