Question

Let $P,Q,R$ be three points on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and $P^{\prime },Q^{\prime }$ and $R^{\prime }$ be the corresponding points on the auxiliary circle. Then,
Area of $\Delta$PQR : Area of $\Delta$ P' Q' R' $=$

• A. $a:b$
• B. $b:a$
• C. $a^2:b^2$
• D. $b^2:a^2$

Answer 1

The correct option is A $a:b$

$P\left({\theta }_1\right),Q\left({\theta }_2\right),R\left({\theta }_3\right)$ , $P^{\prime }\left({\theta }_1\right)=(a\cos {\theta }_1,a\sin {\theta }_1)$

$Q^{\prime }\left({\theta }_2\right)=(a\cos {\theta }_2,a\sin {\theta }_2)$

$R^{\prime }\left({\theta }_3\right)=(a\cos {\theta }_3,a\sin {\theta }_3)$

$△PQR=\left|\begin{array}{ccc}1& 1& 1\\ a\cos {\theta }_1& a\cos {\theta }_2& a\cos {\theta }_3\\ b\sin {\theta }_1& b\sin {\theta }_2& b\sin {\theta }_3\end{array}\right|$

$△PQR=ab\left|\begin{array}{ccc}1& 1& 1\\ \cos {\theta }_1& \cos {\theta }_2& \cos {\theta }_3\\ \sin {\theta }_1& \sin {\theta }_2& \sin {\theta }_3\end{array}\right|$

$△P^{\prime }{Q}^{\prime }{R}^{\prime }=ab\left|\begin{array}{ccc}1& 1& 1\\ a\cos {\theta }_1& a\cos {\theta }_2& a\cos {\theta }_3\\ a\sin {\theta }_1& a\sin {\theta }_2& a\sin {\theta }_3\end{array}\right|=\frac{a^2}{ab}△PQR$

$\frac{△P^{\prime }{Q}^{\prime }R}{△PQR}=\frac{a}{b}$