# Geometrically interpreting the answer to a vector calculus question involving tangent line segments to ellipses.

Let E be an ellipse centered at the origin on the x, y plane with major radius b and minor radius a. The length of the shortest line segment tangent to E that begins on the x-axis and ends on the y-axis is a+b. This can be shown using Lagrange multipliers. This answer is very simple and leads us to ask the following question:

Can you give a geometric reason for why the length is a+b?

This was originally asked to me by Frank Jones a few years ago.

There is a geometric way to show that $n$-gon circumscribed around an ellipse has minimal perimeter if it is inscribed in a confocal ellipse. From Poncelet porism (and generalization of optical property) it follows that we have continuous family of "minimal" polygons.
If we know it, then it is easy to understand that the circumscribed rhomb (from your question) and the circumscribed rectangular (with perimeter $4(a+b)$) are minimal polygons. So, side of the rhomb equals $a+b$.
• Poncelet's porism is quite elementary to see in this case. Consider the case when $a=b=1$ of a circle. Then the two rectangles are the same, and are inscribed in a circle of radius $\sqrt{2}$. Poncelet's porism just says that there is a 1-parameter family of circumscribing/inscribing squares, in this case related by rotation. The affine transformation $(ax, by)$ gives a projective transformation taking this picture to the ellipse with major and minor axes $a,b$. Then the rotations are sent projectively to a 1-parameter family of projective transformations fixing both ellipses. – Ian Agol Mar 1 '10 at 17:51