Paul is back with another elliptical quandary. Suppose you are constrained to a circular track, which we will model as the unit circle in the plane, and there is something smelly at one point on the track, say where it intersects the -axis at the point . If you want to get as far away as possible from the smell, clearly you should run to the diametrically opposite point on the track, namely the point . Now suppose you change the shape of the track by stretching it in the -direction into an ellipse. If you stretch it far enough, then the diametrically opposite point will no longer be the farthest away: as shown in Figure 1, when you have stretched it by a factor of four, most of the ellipse lies more than two units away (the portion shown in green).
What is the most you can stretch the track so that the opposite point remains the farthest place to hide from the smell? (Paul originally asked this question in terms of the eccentricity of the elliptical track: if the circle is stretched by a factor of in the -direction, then the eccentricity of the resulting ellipse is . You may phrase your answer in terms of either the eccentricity or the stretch factor .)