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 $x$-axis at the point $(1,0)$. 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 $(-1,0)$. Now suppose you change the shape of the track by stretching it in the $y$-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 $(-1,0)$ 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 $a>1$ in the $y$-direction, then the eccentricity of the resulting ellipse is $e=\sqrt{1-1{\textstyle \phantom{\rule{-0.167em}{0ex}}}/{\textstyle \phantom{\rule{-0.167em}{0ex}}}{a}^{2}}$. You may phrase your answer in terms of either the eccentricity $e$ or the stretch factor $a$.)