If $l_1$ and $l_2$ are the lengths of the perpendiculars drawn from a point on the hyperbola $5 x^2-4 y^2-20=0$ to its asymptotes, then $\frac{l_1^2 l_2^2}{100}=$

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Aicharya · Accepted Answer

$S \equiv \frac{x^2}{4}-\frac{y^2}{5}=1$ It's asymptotes are $\sqrt{5} x-2 y=0$ or $\sqrt{5} x+2 y=0$ $\begin{aligned} & l_1=\frac{\left|\sqrt{5} x_0-2 y_0ight|}{3} 	ext { and } l_2=\frac{\left|\sqrt{5} x_0+2 y_0ight|}{3} \ & l_1 l_2=\frac{5 x_0^2-4 y_0^2}{9}=\frac{20}{9} \Rightarrow \frac{l_1^2 l_2^2}{100}=\frac{4}{81} \end{aligned}$

If $l_1$ and $l_2$ are the lengths of the perpendiculars drawn from a point on the hyperbola $5 x^2-4…

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