# A problem by Sudeep Salgia

Level pending

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:
$$a = 5$$
$$b = 3$$

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