Question
Question: The tangent at a point P on the hyperbola \(\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1\) meets on...
The tangent at a point P on the hyperbola a2x2−b2y2=1 meets one of its directrices in F. If PF subtends an angle θ at the corresponding focus, then θ equals
A
π/4
B
π/2
C
3π/4
D
π
Answer
π/2
Explanation
Solution
Let the directrix be x = a/e and the focus be S(ae, 0). Let P(asecθ, b tanθ) be any point on the curve.
Equation of tangent at ‘P’ is axsecθ−bytanθ=1.
Let ‘F’ be the intersection point of the tangent and the directrix, so that
F ≡ (a/e,etanθb(secθ−e)) ⇒ mSF = −atanθ(e2−1)b(secθ−e), mPS = a(secθ−e)btanθ
⇒mSF . mPS = –1.