Question

For the ellipse $x^2+4y^2=9,$ which of the following option(s) is/are correct

• A. the eccentricity is $\frac{1}{2}$
• B. the length of latus-rectum is $\frac{3}{2}$
• C. one focus is at $(3\sqrt{3},0)$
• D. one directrix is $x=-2\sqrt{3}$

Answer 1

Answer: (B), (D)

The correct options are
B the length of latus-rectum is $\frac{3}{2}$
D one directrix is $x=-2\sqrt{3}$

$x^2+4y^2=9$
$\Rightarrow \frac{x^2}{9}+\frac{y^2}{\frac{9}{4}}=1$
Here, $a>b$
$\therefore e=\sqrt{1-\frac{b^2}{a^2}}$
$\Rightarrow e=\sqrt{1-\frac{\frac{9}{4}}{9}}$
$\Rightarrow e=\frac{\sqrt{3}}{2}$
Length of latus rectum $=\frac{2b^2}{a}$
$=\frac{2\times \frac{9}{4}}{3}$
$=\frac{3}{2}$ units

Coordinates of foci are
$(\pm ae,0)\equiv (\pm \frac{3\sqrt{3}}{2},0)$

Equation of directix are
$x=\pm \frac{a}{e}$
$\Rightarrow x=\pm 2\sqrt{3}$