First slide

Hyperbola in conic sections

Question

The coordinates of a point common to a directrix and an asymptote of the hyperbola $\frac{x^{2}}{25} - \frac{y^{2}}{16} = 1$ are

Moderate

A

$(\frac{25}{\sqrt{41}}, \frac{20}{3})$
B

$(\frac{20}{\sqrt{41}}, \frac{- 25}{\sqrt{41}})$
C

$(\frac{25}{3}, \frac{20}{3})$
D

$(\frac{- 25}{\sqrt{41}}, \frac{20}{\sqrt{41}})$

Solution

$e^{2} = \frac{25 + 16}{25} \Rightarrow e = \frac{\sqrt{41}}{5}$
Directrix is $x = \pm \frac{25}{\sqrt{41}} (x = \pm \frac{a}{e})$
Asymptotes are $\frac{x^{2}}{25} - \frac{y^{2}}{16} = 0$
So the required coordinates are $(- \frac{25}{\sqrt{41}}, \frac{20}{\sqrt{41}})$

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