Chapter 12.2, Problem 24E

To determine

To prove:

The set of points $C:=\left\{\left(x,y\right):x^2+y^2=r^2\right\}$ is mapped into the set of points $E:=\left\{\left(u,\upsilon \right):\left(a^2+c^2\right)u^2+\left(2ab+2cd\right)u\upsilon +\left(b^2+d^2\right){\upsilon }^2=r^2\right\}$ using the change of variables $x=au+b\upsilon ,y=cu+d\upsilon$ and verify $E$ is an ellipse in the $u\upsilon$ plane.

Here, $ad-bc\ne 0$.