We’ve featured a few Dilemmas from PMP participants about the properties of ellipses. But it turns out that they are a perennial source of interesting questions and problems for the entire mathematical community. For example, six years ago, when the Problem Warden was the Playground editor for the Mathematical Association of America, he wrote down this problem (but it hasn’t been published until now).
- Is the ellipse shown to the right the only one tangent to the x-axis at (0, 2) and the y-axis at (1, 0)? If not, can you find them all, and determine the set of all possible foci (or the set of all possible centers, if you prefer), of such ellipses?
- More generally, if you specify two non-parallel lines and a point on each line, is there always an ellipse that is simultaneously tangent to the respective lines at the two points? When there is at least one, can you find them all and determine the set of possible foci or centers?