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

Question

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

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

Monjur Alahi

5 views

2025-01-04 04:42

1 Answers

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Salim Ahmed Kawsar · Accepted Answer · 2023-02-05T15:29:48+06:00

Correct Answer - B
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`