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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 x2a2y2b2=1\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 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 xsecθaytanθb=1\frac{x\sec\theta}{a} - \frac{y\tan\theta}{b} = 1.

Let ‘F’ be the intersection point of the tangent and the directrix, so that

F ≡ (a/e,b(secθe)etanθ)\left( a/e,\frac{b\left( \sec\theta - e \right)}{e\tan\theta} \right) ⇒ mSF = b(secθe)atanθ(e21)\frac{b\left( \sec\theta - e \right)}{- a\tan\theta\left( e^{2} - 1 \right)}, mPS = btanθa(secθe)\frac{b\tan\theta}{a\left( \sec\theta - e \right)}

⇒mSF . mPS = –1.