Question

If the equation of the hyperbola is $\frac{y^{2}}{9}-\frac{x^{2}}{27}=1$, then
I. the coordinates of the foci are $(0, \pm 6)$
II. the length of the latus rectum is 18 units.
III the eccentricity is $\frac{4}{5}$.

• A. Only I is true.
• B. Only II is true.
• C. Only I and II is true.
• D. Only II and III is true.

Answer 1

Answer: (C)

Comparing the equation $\frac{y^{2}}{9}-\frac{x^{2}}{27}=1$ with the
standard equation. We have, $a=3, b=3 \sqrt{3}$
and $c=\sqrt{a^{2}+b^{2}}=\sqrt{9+27}=\sqrt{36}=6$
Therefore, the coordinates of the foci are $(0, \pm 6)$ and that of vertices are $(0, \pm 3)$. Also, the eccentricity
$e=\frac{c}{a}=\frac{5}{4}$ and the length of latus rectum $=\frac{2 b^{2}}{a}$
$=\frac{2(3 \sqrt{3})^{2}}{3}=\frac{54}{3}=18$ units