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

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

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

$(25 / 3, 20 / 3)$
D

$(- 25 / 3, - 20 / 3)$

Solution

Equation of the directrices of the given hyperbola
$\frac{x^{2}}{25} - \frac{y^{2}}{16} = 1 are x = \pm \frac{5 \times 5}{\sqrt{25 + 16}} = \pm \frac{25}{\sqrt{41}} - - - (i)$
Equations of the asymptotes of the given hyperbola are
$\frac{x^{2}}{25} - \frac{y^{2}}{16} = 0 - - - (ii)$
The points of intersection of Eqs. (i) and (ii) are
$(\pm 25 / \sqrt{41}, \pm 20 / \sqrt{41})$

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