Question

The eccentricity of the ellipse 4x² + 9y² + 8x + 36y + 4 = 0 is

विकल्प

• \[\frac{5}{6}\]

   

• \[\frac{3}{5}\]

   

• \[\frac{\sqrt{2}}{3}\]

   

• \[\frac{\sqrt{5}}{3}\]

   

Answer 1

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