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Question

If the chord through the points whose eccentric angles are α and β on the ellipse x2a2+y2b2=1 passes through the focus (ae,0), then the value of tan α2 tan β2=

A
e+1e1
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B
e1e+1
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C
e+1e2
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D
None
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Solution

The correct option is D e1e+1
Equation of chord to the ellipse with eccentric angle α and β is given by,
xacosα+β2+ybsinα+β2=cosαβ2
Given it passes through (ae,0)
ecosα+β2=cosαβ2
e(cosα2cosβ2sinα2sinβ2)=cosα2cosβ2+sinα2sinβ2
e(1tanα2tanβ2)=1+tanα2tanβ2
tanα2tanβ2=e1e+1

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