Question

For the ellipse x² + 4y² = 9

विकल्प

• the eccentricity is 1/2

• the latus-rectum is 3/2

• a focus is \[\left( 3\sqrt{3}, 0 \right)\]

   

•  a directrix is x = \[- 2\sqrt{3}\]

   

Answer 1

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