Q. The equation of the hyperbola whose vertices are at $(5,0)$ and $( - 5,\ 0)$ and one of the directrices is $x = \frac{25}{7},$ is
A) $\frac{x^{2}}{25} - \frac{y^{2}}{24} = 1$
B) $\frac{x^{2}}{24} - \frac{y^{2}}{25} = 1$
C) $\frac{x^{2}}{16} - \frac{y^{2}}{25} = 1$
D) $\frac{x^{2}}{25} - \frac{y^{2}}{16} = 1$
Explanation
(a)
Given vertices are $(5,\ 0),( - 5,)$
$\therefore a = 5$
Also, one of the directrix let $x = \frac{a}{e}$ is
Given as $x = \frac{25}{7} \Rightarrow \ e = \frac{7}{5}$
$\therefore b^{2} = a^{2}\left( e^{2} - 1 \right) = 25\left( \frac{49}{25} - 1\ \right) = 24$
Equation hyperbola is $\frac{x^{2}}{25} - \frac{y^{2}}{24} = 1$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.