The applet below shows vertical lines at [math]x=\pm a.[/math] You can change the locations of these lines by moving the blue point. We want to see the envelope of lines joining [math]\left(a,y\right)[/math] to [math]\left(-a,\frac{1}{y}\right)[/math] for all nonzero y-values. In the applet, the blue lines start from evenly-spaced points on the line [math]x=a,[/math] and the red lines start from evenly-spaced points on [math]x=-a.[/math] As you can see, the envelope of these lines is an ellipse (and a circle when [math]a=1[/math]).