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Let P be any point on a directrix of an ...

Let `P` be any point on a directrix of an ellipse of eccentricity `e, S` be the corresponding focus, and `C` the center of the ellipse. The line `PC` meets the ellipse at `A`. The angle between at `A` is `alpha`. Then `alpha` is equal to

A

a)`tan^(-1)e`

B

b)`pi//2`

C

c)`tan^(-1)(1-e^(2))`

D

d)none of these

Text Solution

Verified by Experts

The correct Answer is:
B

Let the ellipse be
`(x^(2))/(a^(2))+(y^(2))/(b^(2))=1`
and let `A-=(a cos theta, b sin theta)`
The equation of AC will be
`y=(b)/(a) tan thetax`
Solving with x=a/e, we get
`P-=((a)/(e)(b)/(e) tan theta)`
Slop of tangent at `A=-(b)/(a tan theta)`
Slope of `PS=((b)/(e) tantheta)/((a)/(e)-ae)=(b tantheta)/(a(1-e^(2)))=(a)/(b)tan theta`
So, `alpha=(pi)/(2)`
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