Question

The equation of ellipse whose major axis is along the direction of x-axis, eccentricity is $e=2/3$

• A. $36x^2+20y^2=405$
• B. $20x^2+36y^2=405$
• C. $30x^2+22y^2=411$
• D. $22x^2+32y^2=409$

Answer 1

The correct option is B $20x^2+36y^2=405$

$\begin{array}{c}e=\frac{2}{3}=\sqrt{\frac{a^2-b^2}{c}}\\ \Rightarrow {\left(\frac{2}{3}\right)}^2=\frac{a^2-b^2}{a^2}\\ \Rightarrow \frac{4a^2}{a}=a^2-b^2\\ \Rightarrow b^2=a^2-4a^2=\frac{5a^2}{9}\to \left(i\right)\end{array}$

Equation of Ellipse are

$\begin{array}{c}\Rightarrow \frac{x^2}{a^2}+\frac{y^2}{b^2}=1\\ \Rightarrow \frac{x^2}{a^2}+\frac{9y^2}{5a^2}=1\to \left(ii\right)\\ Put{a}^2=\frac{405}{20}\left(Fromoptioninequation\left(i\right)\right)\\ Then,b^2=\frac{405}{360}\end{array}$

Hence, equation of ellipse is

$\Rightarrow \frac{x^2}{\left(\frac{405}{20}\right)}+\frac{y^2}{\left(\frac{405}{36}\right)}=-1,20x^2+36y^2=405$