How to Change an equation into Ellipse Form

Question

I know how to arrange a normal equation into an ellipse form, but this one is slightly different. $x^2+2xy+5y^2=1$ Any help with this would be greatly appreciated. Thanks

Answer 1

$$1=x^2+2xy+5y^2=(x+y)^2+4y^2$$

so making the variables change $\;\begin{cases}x'=x+y\\y'=y\end{cases}\iff\begin{cases}x=x'-y'\\y=y'\end{cases}\;$

you get the quadratic (ellipse) $\;x'^2+4y'^2=1\;$

Answer 2

Hint

The $xy$ term appears because of the rotation. So, use the standard change of variables $$x=X\cos(t)+Y\sin(t)\qquad , \qquad y=X \sin(t)+Y\cos(t)$$ Replace in $x^2+2xy+5y^2$ and expand. What you need is to cancel the coefficient of the $XY$ terms; before any simplification, this coefficient should be $$2 \cos ^2(t)-8 \sin (t) \cos (t)-2 \sin ^2(t)$$ but this expression simplifies to $$2 (\cos (2 t)-2 \sin (2 t))$$ Setting it to zero will give $t$.

I am sure that you can take it from here.