Consider the matrix $A = \begin{pmatrix} 1 & 1 \\ -1/2 & -2 \end{pmatrix}.$
How do you find the ellipse equation in terms of $x$ and $y?$
I have $X = x+y, \;Y=-\frac{x}{2}-2y.$
Then do we assume $\;X^2 + Y^2 = 1\;$ and $\;X^2 = \frac{x^2}{p^2}, Y^2 = \frac{y^2}{q^2}?$
Why?
$$(x+y)^2+\left( \frac{x}{2}+2y \right)^2=1 \implies \frac{5x^2}{4}+4xy+5y^2=1$$