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 
A$\frac{a}{b}$
B$\frac{b}{a}$
C$\frac{a^2}{b^2}$
D$1$
Answer & Solution
Correct answer: B. $\frac{b}{a}$
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.
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