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

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 $\frac{9}{a^{2}}+\frac{1}{b^{2}}=1$
$\frac{9}{a^{2}}+\frac{5}{3 a^{2}}=1$
$\Rightarrow 27+5=3 a^{2}=32$
$\Rightarrow a^{2}=\frac{32}{3}, b^{2}=\frac{3}{5} \times \frac{32}{3}=\frac{32}{5}$
Required equation of the ellipse is $3 x^{2}+5 y^{2}=32$