this solution isn't polar
we have $Ax^2+Bxy+Cy^2+Dx+Ey+F=0$ as our conic section equation which is rotated.
if $B^2-4AC<0$, then we have an ellipse (If $A=C$ and $B=0$, it is a circle).
if $B^2-4AC=0$, then we have a parabola.
if $B^2-4AC>0$, then we have a hyperbola (If $A+C=0$, then we have a rectangular hyperbola).
our goal is to find the canonical equation of the rotated ellipse
first, we should calculate the Eigenvalues of the following matrix:
for finding the Eigenvalues, we should rewrite the matrix as the following and then find its determinant; then find $\lambda$ (note that you'll find two lambdas) when $|\alpha_2|=0$ ($|\alpha_2|$ is the determinant of the matrix).
now, we should find the determinant of the following matrix:
now, find the canonical equation using the following formula:
if you don't know how to find those determinants, please tell me in the comments.