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Question

If the chord through the points whose eccentric angles are θ and ϕ on the ellipse x225+y29=1 passes through a focus, then possible value(s) of tan(θ2)tan(ϕ2) is/are

A
19
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B
9
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C
19
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D
9
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Solution

The correct options are
B 9
C 19
The equation of the line joining θ and ϕ is
x5cos(θ+ϕ2)+y3sin(θ+ϕ2)=cos(θϕ2)

If it passes through a focus (ae,0)(4,0), then
45cos(θ+ϕ2)=cos(θϕ2)
45=cos(θϕ2)cos(θ+ϕ2)

By applying componendo and dividendo rule, we get
4+545=cos(θϕ2)+cos(θ+ϕ2)cos(θϕ2)cos(θ+ϕ2)
91=2cos(ϕ2)cos(θ2)2sin(ϕ2)sin(θ2)
tan(θ2)tan(ϕ2)=19

If it passes through another focus (4,0)
then (by applying above process again)
tan(θ2)tan(ϕ2)=9

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