Question

What will be the equation of the standard hyperbola where foci are $(0,\pm 10)$ and the length of the latus rectum is $30$ ?

• A. $3y^2-x^2=75$
• B. $y^2-x^2=25$
• C. $3y^2-x^2=25$
• D. $y^2-x^2=75$

Answer 1

Answer: (A)

We have, foci $=(0,\pm be)=0,\pm 10)$
and latus rectum $=30=2a^2/b$
$\Rightarrow a^2=15b$
i.e., $e=\sqrt{1+\frac{15b}{b^2}}$
$\Rightarrow e^2=1+\frac{15e}{10}$
$\Rightarrow 2e^2-3e-2=0$
$\Rightarrow (e-2)(2e+1)=0$
$\Rightarrow e=2$ or $-\frac{1}{2}$
If $e=2$, then $b=\frac{10}{2}=5$
$\Rightarrow a^2=15\times 5=75$
Hence, equation of hyperbola is
$\frac{x^2}{75}-\frac{y^2}{25}=-1$
$\Rightarrow x^2-3y^2=-75$ or
$3y^2-x^2=75$