Question

For the ellipse $x^2+4y^2=9$

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

Answer 1

The correct option is B the latus-rectum is 3/2


the latus-rectum is 3/2
$x^2+4y^2=9$
$\Rightarrow \frac{x^2}{9}+\frac{y^2}{\frac{9}{4}}=1$
Here, a>b
$\therefore e=\sqrt{1-\frac{a^2}{b^2}}$
$\Rightarrow e=\sqrt{1-\frac{9}{4}}$
$\Rightarrow e=\sqrt{1-\frac{1}{4}}$
$\Rightarrow e=\frac{\sqrt{3}}{2}$
Latus rectum = $\frac{2b^2}{a}$
$\frac{2\times \frac{9}{4}}{a}=\frac{3}{2}$
Focus= $(\pm ae,0)$
=$(\frac{3\sqrt{3}}{2},0)$
Directrix $x=\pm \frac{a}{e}$
$=\pm \frac{3}{\frac{\sqrt{3}}{2}}$
$=2\sqrt{3}$