A problem by Sudeep Salgia

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:
\(a = 5\)
\(b = 3\)


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