Question

The eccentricity of the ellipse $4x^2+9y^2+8x+36y+4=0$ is

• A. $\frac{5}{6}$
  
• B. $\frac{3}{5}$
  
• C. $\frac{\sqrt{2}}{3}$
  
• D. $\frac{\sqrt{5}}{3}$

Answer 1

The correct option is D $\frac{\sqrt{5}}{3}$

$4x^2+9y^2+8x+36y+4=0$
$4(x^2+2x)+9(y^2+4y)=-4$
$\Rightarrow (x^2+2x+1)+9(y^2+4y+4)=-4+4+36$
$\Rightarrow 4(x-1{)}^2+9(y+2{)}^2=36$
$\Rightarrow \frac{4(x+1{)}^2}{36}+\frac{9(y+2{)}^2}{36}=1$
$\Rightarrow \frac{(x+1{)}^2}{9}+\frac{(y+2{)}^2}{4}=1$
Comparing it with $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,$ we get:
a=3 and b=2
So, the major and the minor axes of the ellipse are along the x-axis and y-axis, respectively
Now, $e=\sqrt{1-\frac{b^2}{a^2}}$
$\Rightarrow e=\sqrt{1-\frac{4}{9}}$
$\Rightarrow e=\sqrt{\frac{5}{9}}$
$\Rightarrow e=\frac{\sqrt{5}}{3}$