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Parallel Axis Theorem

The tangent a...

Question

The tangent at a point $P$ on the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ meets one of the directrix in $F$ . If $P F$ subtends an angle $θ$ at the corresponding focus, the $θ$ equals :

\frac{π}{4}

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\frac{π}{2}

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\frac{3 π}{4}

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π

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Solution

The correct option is B $\frac{π}{2}$
Let the directrix be $x = \frac{a}{e}$ and focus be $S (a e, 0)$ .
Let $P (a sec θ, b tan θ)$ be any point on the curve.
Equation of tangent at $P$ is $\frac{x sec θ}{a} - \frac{y tan θ}{b} = 1$
Let $F$ be the intersection of tangent and the directrix, then
$F = (\frac{a}{e}, \frac{b (sec θ - e)}{e tan θ})$
$\Rightarrow m_{S F} = \frac{b (sec θ - e)}{- e tan θ (a^{2} - 1)}, m_{P S} = \frac{b tan θ}{a (sec θ - e)} \Rightarrow m_{S F} \times m_{P S} = - 1$

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