Question

Write the centre and eccentricity of the ellipse 3x² + 4y² − 6x + 8y − 5 = 0. 

Answer 1

\[3 x^2 - 6x + 4 y^2 + 8y - 5 = 0\]
\[ \Rightarrow 3( x^2 - 2x) + 4( y^2 + 2y) = 5\]
\[ \Rightarrow 3( x^2 - 2x + 1) + 4( y^2 + 2y + 1) = 5 + 3 + 4\]
\[ \Rightarrow 3(x - 1 )^2 + 4(y + 1 )^2 = 12\]
\[ \Rightarrow \frac{3(x - 1 )^2}{12} + \frac{4(y + 1 )^2}{12} = 1\]
\[ \Rightarrow \frac{(x - 1 )^2}{4} + \frac{(y + 1 )^2}{3} = 1\]
\[\text{ Compairing it with }\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, \text{ we get }: \]
\[a = 2 \text{ and } b = \sqrt{3}\]
\[\text{ Here }, a > b, \text{ so the major and the minor axes of the ellipse are along the x - axis and y - axis, respectively } . \]
\[\text{ Now }, e = \sqrt{1 - \frac{b^2}{a^2}}\]
\[ \Rightarrow e = \sqrt{1 - \frac{3}{4}}\]
\[ \Rightarrow e = \sqrt{\frac{1}{4}}\]
\[ \therefore e = \frac{1}{2} \text{ and centre } =\left( 1, - 1 \right)\]