Coordinate change

Question

I am working on a problem that asks me to find the equation for the ellipse with foci $(1,1)$ and $(-1,-1)$ and passing the point $(2,2)$. So I went ahead and did the coordinate change, in which case $\begin{bmatrix} u \\ v \end{bmatrix} = \begin{bmatrix} 1 & 1 \\ -1 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}$ (It seems that this is the problematic step since I used the standard rotation matrix for checking and it gave me the correct answer). Then the equation for the ellipse in $uv$-coordinate is $\dfrac{u^2}{4} + \dfrac{v^2}{3} = 1$, or $3u^2 + 4v^2 = 12$. Now if I use the previous matrix to substitute $u$ and $v$ in terms of $x$ and $y$, I would obtain $7x^2 - 2xy + 7y^2 = 12$. But apparently the correct equation should be $7x^2 - 2xy + 7y^2 = 48$. What exactly is wrong with this calculation?

Answer 1

I know what went wrong... The actual change-of-basis matrix should be $\begin{bmatrix} 1/2 & 1/2 \\ -1/2 & 1/2 \end{bmatrix}$.