Chapter 10

Let \(C\) be the curve of intersection of a right circular cylinder and a plane making an angle \(\phi(0<\phi<\pi / 2)\) with the axis of the cylinder. Show that \(C\) is an ellipse.

Using the same axes, draw the conics \(y=\) \(\pm\left(a x^{2}+1\right)^{1 / 2}\) for \(-2 \leq x \leq 2\) and \(-2 \leq y \leq 2\) using \(a=\) \(-2,-1,-0.5,-0.1,0,0.1,0.6,1 .\) Make a conjecture about how the shape of the figure depends on \(a\).