Question

Find the centre, the lengths of the axes, eccentricity, foci of the following ellipse: 

 x² + 4y² − 4x + 24y + 31 = 0 

Answer 1

\[ x^2 + 4 y^2 - 4x + 24y + 31 = 0\]
\[ \Rightarrow \left( x^2 - 4x \right) + 4\left( y^2 + 6y \right) = - 31\]
\[ \Rightarrow \left( x^2 - 4x + 4 \right) + 4\left( y^2 + 6y + 9 \right) = - 31 + 36 + 4\]
\[ \Rightarrow \left( x - 2 \right)^2 + 4 \left( y + 3 \right)^2 = 9\]
\[ \Rightarrow \frac{\left( x - 2 \right)^2}{9} + \frac{\left( y + 3 \right)^2}{\frac{9}{4}} = 9\]
\[\text{ Here }, x_1 = 2 \text{ and } y_1 = - 3 \]
\[\text{ Also }, a = 3 \text{ and } b = \frac{3}{2}\]
\[\text{ Centre }=\left( x_1 , y_1 \right)=\left( 2, - 3 \right)\]
\[\text{ Major axis }=2a\]
\[ = 2 \times 3\]
\[ = 6\]
\[\text{ Minor axis }=2b\]
\[ = 2 \times \frac{3}{2}\]
\[ = 3\]
\[e = \sqrt{1 - \frac{b^2}{a^2}}\]
\[ \Rightarrow e = \sqrt{1 - \frac{\frac{9}{4}}{9}}\]
\[ \Rightarrow e = \frac{\sqrt{3}}{2}\]
\[\text{ Foci } = \left( x_1 \pm ae, y_1 \right)\]
\[ = \left( 2 \pm \frac{3\sqrt{3}}{2}, - 3 \right)\]