Question

Let $P,Q,R$ be three points on the ellipse $x^2+4y^2=4$ and $P^{\prime },Q^{\prime },R^{\prime }$ be the corresponding points on the auxiliary circle. Then Area of $△P^{\prime }{Q}^{\prime }{R}^{\prime }$ $:$ Area of $△PQR$ is

Answer 1

We know that,

The ratio of area of any triangle $PQR$ inscribed in the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and that of triangle formed by corresponding points on the auxiliary circle is $\frac{b}{a}.$

Given ellipse is $x^2+4y^2=4$

$\Rightarrow \frac{x^2}{2^2}+\frac{y^2}{1^2}=1$

Lets compare with $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

$a=2$ and $b=1$

The ratio of area of any triangle $PQR$ inscribed in the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and that of triangle formed by corresponding points on the auxiliary circle is $\frac{b}{a}.$

$\frac{\text{ar}\left(△PQR\right)}{\text{ar}(△P^{\prime }{Q}^{\prime }R{)}^{\prime }}=\frac{b}{a}=\frac{1}{2}$

$\Rightarrow \frac{\text{ar}\left(△P^{\prime }{Q}^{\prime }{R}^{\prime }\right)}{\text{ar}\left(△PQR\right)}=\frac{a}{b}=\frac{2}{1}=2$