I need to Find the area of conic section given by:
$5x^2 - 6xy + 5y^2 = 8$, Using substitution $x = u+ v$ and $y = u-v$ we get :
$5\left(u+v\right)^2 - 6\left(u^2 - v^2\right) + 5\left(u-v\right)^2 = 8$
which simplifies to :
$4u^2 + 16v^2 = 8$
thus, this is an equation of ellipse, $\dfrac{u^2}{2} + \dfrac{v^2}{1/2} = 1$,
Area is given by : $\pi a b = \pi \sqrt{2} \sqrt{1/2} = \pi$
Is My Solution to given problem correct ?
Thank you
[Adding this answer in order that yet another question that was answered in a comment not remain formally unanswered.]
As CY Aries points out in a comment, your transformation isn’t area-preserving. You must either rescale the area computed from the transformed equation by multiplying by the Jacobian determinant of the transformation (as you noted), or use an area-preserving transformation in the first place. The one you used originally is a similarity, so it’s easily adjusted to preserve areas: normalize it to $x={u+v\over\sqrt2}$, $y={u-v\over\sqrt2}$. Either way you’ll end up with twice the area that you computed with your original transformation.