The tangent at a point `P` on the hyperbola `(x^2)/(a^2)-(y^2)/(b^2)=1` meets one of the directrix at `Fdot` If `P F` subtends an angle `theta` at the corresponding focus, then `theta=` `pi/4` (b) `pi/2` (c) `(3pi)/4` (d) `pi`

Options

A. `pi//4`
B. `pi//2`
C. `3pi//4`
D. `pi`

Answer

B

Explanation

Let the directrix be x = a/e and the focus be S(ae, 0). Let `P(a sec theta, b tan theta)` be any point on the curve.
The equation of tangent at P is
`(x sec theta)/(a)-(y tan theta)/(b)=1`
Let F be the intersection point of the tangent of directrix. The,
`F-=((a)/(e),(b(sec theta-e))/(e tan theta))`
`therefore" "m_(SF)=(b(sectheta-e))/(-a tan theta(e^(2)-1))`
`andm_(PS)=(b tan theta)/(a(sec theta-e))`
`therefore" "m_(SF^(*))m_(PS)=-1`