<s>[INST] How to bring elliptical equation $2x^2+2y^2+3xy-x-y=0$ into canonical form I have this ellipse: $$2x^2+2y^2+3xy-x-y=0$$
Canonical form is $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$
How can I bring my ellipse to that canonical form? It seems like I need some substitution.
 [/INST] Rewrite the equation,
$$\begin{align}
& 2x^2+2y^2+3xy-x-y\\
& = 2(x+y)^2 -xy -(x+y)\\
& = 2(x+y)^2 -\frac14[(x+y)^2-(x-y)^2] -(x+y)\\
& =\frac74 \left(x+y-\frac27\right)^2+\frac14 (x-y)^2 - \frac17=0 \\
\end{align}$$
Then, let $u=\frac 1{\sqrt2}(x+y-\frac27)$ and $u=\frac 1{\sqrt2}(x-y)$ to get the canonical form
$$\frac{u^2}{\left(\frac{\sqrt2}7\right)^2} + \frac{v^2}{\left(\sqrt{\frac{2}7}\right)^2} =1$$
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