For the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, the ratio of the area of any triangle inscribed in the ellipse to the area of the triangle formed by the corresponding points on its auxiliary circle is 

The mapping from the auxiliary circle $x^2+y^2=a^2$ to the ellipse is $(x,y)\mapsto (x,\frac{b}{a}y)$, which scales all areas by the factor $\frac{b}{a}$. Therefore the area of any inscribed triangle on the ellipse is $\frac{b}{a}$ times the area of the corresponding triangle on the auxiliary circle.