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 ?

Under the variable changes $x = \frac{u+v}{\sqrt2}$ and $y = \frac{u-v}{\sqrt2}$, the equation $5x^2 -6xy +5y^2=8$ is of the standard ellipse form
$$\frac{u^2}4+v^2=1$$
with the major and minor axes 2 and 1, respectively. Thus, the area is $2\pi$.
