<s>[INST] Find area of ellipse $5x^2 -6xy +5y^2=8$ 
Find the  area of ellipse whose  equation in the  $xy$- plane  is given by  $5x^2 -6xy +5y^2=8$

My attempt  : I know  that area   of ellipse $ = \pi a b$ ,where $a$ is  semi-major axis and $b$ is  semi minor axis
Now  if we make  matrix $\begin{bmatrix} 5 & -3 \\-3& 5\end{bmatrix}$ Here  eigenvalue  $\lambda_1=  8 ,\lambda_2=2$
That  is area of ellipse$ = \pi \frac{1}{\sqrt\lambda_2} \frac{1}{\sqrt\lambda_1}= \pi \frac{1}{2\sqrt 2}\frac{1}{\sqrt2}$
Is its  correct ?
 [/INST] Given $$5x^2-6xy+5y^2=8$$
Curve represent ellipse whose center is at $(0,0)$ and it is symmetrical about $y=x$ and $y=-x$ line

Put $y=x$ in $5x^2-6xy+5y^2=8$, getting $=\pm \sqrt{2}.$
So point of intersection of line and ellipse is $(\sqrt{2},\sqrt{2})$ or $(-\sqrt{2},-\sqrt{2})$
Which is notning but distance length of semi major axis i e $a = 2$
Put $y=-x$ in $5x^2-6xy+5y^2=8$, getting $\displaystyle =\pm \frac{1}{\sqrt{2}}.$
So point of intersection of line and ellipse is $\displaystyle \bigg(\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}}\bigg)$ or $\displaystyle \bigg(-\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}\bigg)$
Which is notning but distance length of semi minor axis i e $b=1$
So we have eccentricity $$e=\sqrt{1-\frac{b^2}{a^2}}=\frac{\sqrt{3}}{2}$$
And area of ellipse is $$\pi ab=2\pi$$
</s>