A problem by Sudeep Salgia

Level 2

Consider a standard ellipse:
x2a2+y2b2=1 \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1
where aa and bb 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. ,
x2+y2=a2 x^2 + y^2 = a^2

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

The points are defined as :
P(θ)=(acosθ,bsinθ) P(\theta) = (a\cos \theta , b\sin \theta)
P(θ)=(acosθ,asinθ) P'(\theta) = (a\cos \theta , a\sin \theta)

Given, the pairs of complimentary points namely (A,AA , A') , (B,BB , B') and (C,CC , C') with parameters α \alpha , β\beta and γ\gamma , i.e., the points on the ellipse are A(α)A(\alpha) , B(β)B(\beta) and C(γ)C(\gamma).

Find the ratio of the area of the triangle ABCABC to that of the triangle ABCA'B'C'.

Details and Assumptions:
a=5a = 5
b=3b = 3

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