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 ?
Best Answer
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 $2\pi$.