Question

What will be the equation of the standard hyperbola where foci are $ (0, ± 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 $= (0, ± be) = 0, ± 10)$
and latus rectum $= 30 = 2a^2 /b$
$\Rightarrow a^2 = 15b$
i.e., $e = \sqrt{1+\frac{15 b}{b^2}}$
$\Rightarrow e^2 = 1 + \frac{15e}{10}$
$\Rightarrow 2e^2 - 3e - 2 = 0$
$\Rightarrow (e - 2)(2e + 1) = 0$
$\Rightarrow e = 2$ 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
$3y^2 - x^2 = 75$