MAA members will likely recognize the following challenge that appeared on the outside of the envelope in the mailing to members or prospective members. I plan on giving this to my AP Calculus students as review for the exam or afterwards. As usual I will modify it for the student, place it in the context of an activity, broken into several parts with some hints. The original problem comes with a helpful diagram, however, unless I scan it, it would be difficult to reproduce. The problem involves a property of a point on an ellipse and requires basic understanding of the parametric form of this curve and some basic calculus and trig. The last part of the activity suggests a possible significance of this property but I'll leave the details to our astute readers.

Consider a standard ellipse, center at (0,0), with major axis of length 2a on the x-axis and minor axis of length 2b.

Let P(x,y) be a generic point on this ellipse with the restriction that P is not one of the endpoints of the major or minor axes. Consider the tangent and normal lines at P. Let P denote the point of intersection of the normal line with the x-axis and Q, the point of intersection of the tangent line with the x-axis.

Prove that (OP)(OQ) = a^{2}-b^{2}, where OP represents the distance between the origin and P and similarly for OQ.

Here is an outline with several parts for the student:

(a) Show that x = acos(t), y = bsin(t), 0<=t<2pi,>2-b^{2})/a)cos(t).

(f) Use (d) and (e) to derive the desired result: (OP)(OQ) = a^{2}-b^{2}

(g) Explain why we did not allow P to be an endpoint of the major or minor axes.

(h) What does the expression a^{2}-b^{2} have to do with the foci of the ellipse? For EXTRA CREDIT, investigate this 'focal' property further.

## Sunday, April 15, 2007

### A Challenge Problem: Ellipses and Tangents and Normals

Posted by Dave Marain at 8:40 AM

Labels: calculus, ellipse, MAA, parametric, tangents, trigonometry

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