Problem:
A plane intersects a right circular cylinder of radius forming an ellipse. If the major axis of the ellipse is longer than the minor axis, the length of the major axis is
Answer Choices:
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Solution:
We claim that the minor axis has length , in which case the major axis has length . To prove the claim, we may assume that the equation of the cylinder is and that the intersection of the plane and the axis of the cylinder is . By symmetry, is also the center of the ellipse. Furthermore, the line of intersection of the given plane with the plane contains two points on the ellipse (and the cylinder), and they have . Finally, each "radius" of the ellipse extends from ( ) to some point on the cylinder and thus has length . This has minimum value when , which is obtained for the two diametrically opposed points described earlier. Thus the minor axis does indeed have length .
Note. For those with good spatial intuition, it should be evident that the minor axis of the ellipse is a diameter of the cylinder.