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P is any point lying on the ellipse (x^(...

P is any point lying on the ellipse `(x^(2))/(a^(2))+(y^(2))/(b^(2))=1(agtb)` whose foci are `S` and `S'`. If `anglePSS'=alpha and anglePS'S=beta`, then the value of `tan.(alpha)/(2)tan.(beta)/(2)` is

A

`(1+e)/(1-e)`

B

`(1+e^(2))/(1-e^(2))`

C

`(1-e)/(1+e)`

D

1

Text Solution

Verified by Experts

The correct Answer is:
C

SP+S''P=2a
In `DeltaSPS''`
Perimeter, `2s=SP+S'P+SS'=2a+2ae=2a(1+e)`
Now, `tan.(alpha)/(2)tan.(beta)/(2)=sqrt(((s-b)(s-c))/(s(s-c)))sqrt(((s-a)(s-c))/(s(s-b)))`
`s(-c)/(s)=(2s-2c)/(2s)=(2a(1+e)-2c)/(2a(1+e))`
`(2a+2ae-4ae)/(2a(1+e))=(2a(1-e))/(2ae(1+e))`
`:.tan.(alpha)/(2)tan.(beta)/(2)=(1-e)/(1+e)`
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