Level
2

Consider a standard ellipse:

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

where $a$ and $b$ are the lengths of semi-major and semi-minor axes respectively.

Now, consider a circle concentric with the ellipse and the radius equal to the length of the semi-major axis of the ellipse, i.e. ,

$x^2 + y^2 = a^2$

A set of complimentary points are defined on these two conics, $P$ and $P'$ with a parameter, $\theta$. Point $P$ is lies on the ellipse and $P'$ on the circle.

The points are defined as :

$P(\theta) = (a\cos \theta , b\sin \theta)$

$P'(\theta) = (a\cos \theta , a\sin \theta)$

Given, the pairs of complimentary points namely ($A , A'$) , ($B , B'$) and ($C , C'$) with parameters $\alpha$ , $\beta$ and $\gamma$ , i.e., the points on the ellipse are $A(\alpha)$ , $B(\beta)$ and $C(\gamma)$.

Find the ratio of the area of the triangle $ABC$ to that of the triangle $A'B'C'$.

**Details and Assumptions:**

$a = 5$

$b = 3$