Hyperbola in conic sections

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Question

$If P Q is a double ordinate of the hyperbola 16 x^{2} - 25 y^{2} = 400 be produced on both$ $sides to meet the asymptotes at R and S, then P R \cdot P S =$

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Solution

$\begin{aligned} Given hyperbola is \\ 16 x^{2} - 25 y^{2} = 400 \end{aligned}$
$\Rightarrow \frac{x^{2}}{25} - \frac{y^{2}}{16} = 1$
$Let P (a \sec θ, b \tan θ) be a point on hyperbola$
$\Rightarrow Q (a \sec θ, - b \tan θ) (Image of P with respect to x -axis)$
$Equation of P Q is x = a \sec θ$
$It intersects with asymptotes \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 0$
$\Rightarrow y = \pm b \sec θ$
$\begin{array}{l} So, R (a \sec θ, b \sec θ), S (a \sec θ, - b \sec θ) \\ ∴ P R \cdot P S = b^{2} (\sec^{2} θ - \tan^{2} θ) \\ = b^{2} \\ = 16 \end{array}$

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Hyperbola in conic sections

Remember concepts with our Masterclasses.

If PQ is a double ordinate of the hyperbola 16x2−25y2=400 be produced on both sides to meet the asymptotes at R and S, then PR⋅PS=

Ready to Test Your Skills?

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$If P Q is a double ordinate of the hyperbola 16 x^{2} - 25 y^{2} = 400 be produced on both$ $sides to meet the asymptotes at R and S, then P R \cdot P S =$