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Question: The product of perpendicular drawn from any point on \(\frac{x^{2}}{9} - \frac{y^{2}}{16}\) = 1 upon...

The product of perpendicular drawn from any point on x29−y216\frac{x^{2}}{9} - \frac{y^{2}}{16} = 1 upon its asymptote is-

A

125144\frac{125}{144}

B

14425\frac{144}{25}

C

25144\frac{25}{144}

D

144125\frac{144}{125}

Answer

14425\frac{144}{25}

Explanation

Solution

Asymptote for x2a2−y2b2\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 are bx – ay = 0 and bx + ay = 0.

Let a point on hyperbola is (a secq, b tanq)

Product of length of perpendicular

= ∣absec⁡θ−abtan⁡θ∣a2+b2\frac{|ab\sec\theta - ab\tan\theta|}{\sqrt{a^{2} + b^{2}}} × ∣absec⁡θ+abtan⁡θ∣a2+b2\frac{|ab\sec\theta + ab\tan\theta|}{\sqrt{a^{2} + b^{2}}}

= a2b2a2+b2\frac{a^{2}b^{2}}{a^{2} + b^{2}} = 9×169+16\frac{9 \times 16}{9 + 16} = 14425\frac{144}{25}