Quandaries
and Queries 

Who is asking: Student Question: thanks! 

Hi Judy, The general feeling here was that the outside rail does not form an ellipse so I looked at an example. Suppose that you have the ellipse
and the track is one unit wide so that the outer rail goes through (3,0), 0,2), (3,0) and (0,2). The ellipse that passes through these points is
I then took the point P ( ^{2}/_{sqrt(5)}, ^{2}/_{sqrt(5)}) on the inside ellipse and calculated the distance to the point Q on the outside ellipse. What is the point Q? I think that you should measure in a direction "perpendicular to the inside ellipse".To do this I used calculus to find the slope of the tangent to the inside ellipse at P (the slope is ^{1}/_{4}) and then wrote the equation of the line through P that is perpendicular to this tangent, ie the line through P with slope 4.This line intersects the outside ellipse at (1.13696, 1.8508). The distance from P to Q is 0.9866 and thus at this point the width of the track is less than 1 unit. Hence the outer rail is not an ellipse. We are curious where this problem came from. Was it an assignment? Penny 

