Derivation of dilation property of ellipses
The rectangular coordinate equation of an ellipse looks a lot like the equation of the unit circle,
Does their similarity tell us something?
Let's take one piece at a time. What if we divide x by a in the equation? Dividing all occurrences of the variable x in an equation by a number has the effect of dilating the figure—stretching it—horizontally by a factor of a. (If a is less than 1, then the "stretching" is actually a "shrinking.") Likewise, a vertical dilation takes place if all occurrences of y are divided by a constant b.
So the equation tells us that any ellipse is the result of two dilations of the unit circle.
Actually, the equation for an ellipse could be derived from the equation
which represent any circle of radius r. If r = a, then the horizontal diameter of the circle coincides with the horizontal axis of the ellipse. We can get the vertical diameter to match with the vertical axis by dilating the circle vertically. To do that we divide the y by b/a, and thus we get the standard equation for an ellipse.
