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

• A. $(\frac{25}{\sqrt{41}},\frac{20}{\sqrt{41}})$
• B. $(\frac{25}{\sqrt{41}},-\frac{20}{\sqrt{41}})$
• C. $(-\frac{25}{\sqrt{41}},\frac{24}{\sqrt{41}})$
• D. $(\frac{25}{\sqrt{41}},\frac{24}{\sqrt{41}})$

Answer 1

Answer: (A), (B)

The correct options are
A $(\frac{25}{\sqrt{41}},\frac{20}{\sqrt{41}})$
B $(\frac{25}{\sqrt{41}},-\frac{20}{\sqrt{41}})$

Equation of the directrices of the given hyperbola are
$\frac{x^2}{25}-\frac{y^2}{16}=1$ are
$x=\pm \frac{5}{\frac{\sqrt{25+16}}{5}}=\frac{\pm 25}{\sqrt{41}}\cdots \left(1\right)$

Equations of asymptotes of the given hyperbola are
$\frac{x^2}{25}-\frac{y^2}{16}=0$
$y=\pm \frac{b}{a}x=\pm \frac{4}{5}x\cdots \left(2\right)$
$\therefore$ Points of intersection of $\left(1\right)$ and $\left(2\right)$ are :
$(\frac{25}{\sqrt{41}},\pm \frac{20}{\sqrt{41}}),(-\frac{25}{\sqrt{41}},\pm \frac{20}{\sqrt{41}})$