Area of ellipse $\frac{x^2}{a^2}+ \frac{y^2}{b^2}+\frac{xy}{ab}-\frac{x}{a}-\frac{y}{b}=0$

Question

Is someone able to help me with this question?

For a>0, b>0, find the area of the area D delimited by the ellipse

$$\frac{x^2}{a^2}+ \frac{y^2}{b^2}+\frac{xy}{ab}-\frac{x}{a}-\frac{y}{b}=0$$

by doing a change of variable: $$ \frac{x}{a}= Au+Bv$$ and $$\frac{y}{b}=Cu+Dv$$

I dont know how to start attacking this problem, any help would be very appreciated. Thanks a lot! By the way,English is not my first langage so sorry if there is any mistakes.

Answer 1

Re-center and rescale the ellipse with $x=a(t+\frac13)$ and $y=b(s+\frac13)$,

$$t^2+ts+s^2 = \frac13$$

Then, rotate with $t=\frac1{\sqrt2}(u+v)$ and $s=\frac1{\sqrt2}(u-v)$,

$$\frac{u^2}{\frac29}+\frac{v^2}{\frac23} = 1$$

Therefore, the area of the ellipse is

$$A= ab\pi \sqrt{\frac29\cdot\frac23}=\frac{2\sqrt3\pi}9ab$$