I've got a problem in which I'm trying to find the area of an ellipse which is given by the intersection of an elliptic cylinder with a plane. Nothing here is parallel to the coordinate axes, which is making it kind of annoying to work with.
The plane is given by the equation $x+ay+a^2z=0$, and the cylinder is given by $(x-a^2z)^2-a(x-a^2z)(y-az)+a^2(y-az)^2=L^2$.
I can think of some complicated ways to do it with integrals, and I'm wondering if there's something simpler that I'm missing. If I could transform into some coordinates in the plane I'm trying to work with, and I knew that the coordinate transformation would preserve areas, then I would be good, because I know how to get the area of an ellipse in the form $Ax^2+Bxy+Cy^2=1$. I'm really not sure how to do this transformation, though, and whether this is even the best way to proceed.
Maybe I should be using Langrange multipliers with two constraints to just obtain the lengths of the semi-major and semi-minor axes? That sounds like a pain, but doable. At least we're centered at the origin.
Thanks in advance for any assistance.
Edit: If I take $u=(x-a^2z)$ and $v=(y-az)$, then my cylinder becomes $u^2-auv+a^2v^2=L^2$, which is pretty great, but then I've basically cut the cylinder with a plane parallel to $z=0$. In that case, my given plane becomes $u+av+3a^2z=0$, and I'm not sure how that's helpful.