Question

Equation of the ellipse whose axes are the axes of coordinates and which passes through the point (-3,1) and has eccentricity $\sqrt{\frac{2}{5}}$ is

• A. $5x^2+3y^2-32=0$
• B. $3x^2+5y^2-32=0$
• C. $5x^2+3y^2-48=0$
• D. $3x^2+5y^2-15=0$

Answer 1

Answer: (B)

Let the equation of the ellipse be $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$
Which will pass through (-3,1) if $\frac{9}{a^2}+\frac{1}{b^2}=1$
and eccentricity $=e=\sqrt{1-\frac{b^2}{a^2}}=\sqrt{\frac{2}{5}}$
$\Rightarrow \frac{2}{5}=1-\frac{b^2}{a^2}$
$\Rightarrow \frac{b^2}{a^2}=\frac{3}{5}$
$\Rightarrow b^2=\frac{3}{5}{a}^2$
Thus $\frac{9}{a^2}+\frac{1}{b^2}=1$
$\frac{9}{a^2}+\frac{5}{3a^2}=1$
$\Rightarrow 27+5=3a^2=32$
$\Rightarrow a^2=\frac{32}{3},b^2=\frac{3}{5}\times \frac{32}{3}=\frac{32}{5}$
Required equation of the ellipse is $3x^2+5y^2=32$