10 tangents, 20 feets!

Geometry Level 3

Let P i P_i and P i ′ P'_i be the feet of perpendiculars drawn from foci S S and S ′ S' on a tangent T i T_i to the ellipse E : x 2 a 2 + y 2 b 2 = 1 E: \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 whose length of major axis is 40 units.

If ∑ i = 1 10 ( S P i ) ( S P i ′ ) = 2560 \ \displaystyle \sum_{i=1}^{10} (SP_i)(SP'_i) = 2560 , then find the value of 100 e 100e .

Details:

' e e ' denotes the eccentricity of the ellipse.


The answer is 60.

Tapas Mazumdar by Tapas Mazumdar , 20, India

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1 solution

Prakhar Bindal
Dec 19, 2016

Simply use the property that product of length of perpendicular from the focii to any tangent is b^2

so b^2 = 256 b = 16 and a = 20

simply e = 3/5

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