Question

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

4x² + y² − 8x + 2y + 1 = 0 

Answer 1

\[ 4 x^2 + y^2 - 8x + 2y + 1 = 0\]
\[ \Rightarrow 4\left( x^2 - 2x \right) + \left( y^2 + 2y \right) = - 1\]
\[ \Rightarrow 4\left( x^2 - 2x + 1 \right) + \left( y^2 + 2y + 1 \right) = - 1 + 4 + 1\]
\[ \Rightarrow 4 \left( x - 1 \right)^2 + \left( y + 1 \right)^2 = 4\]
\[ \Rightarrow \frac{\left( x - 1 \right)^2}{1} + \frac{\left( y + 1 \right)^2}{4} = 1\]
\[\text{ Here }, x_1 = 1 a\text{ and } y_1 = - 1 \]
\[\text{ Also }, a = 1 \text{ and } b = 2\]
\[\text{ Centre }=\left( x_1 , y_1 \right)=\left( 1, - 1 \right)\]
\[\text{ Major axis }=2b\]
\[ = 2 \times 2\]
\[ = 4\]
\[\text{ Minor axis }=2a\]
\[ = 2 \times 1\]
\[ = 2\]
\[e = \sqrt{1 - \frac{a^2}{b^2}}\]
\[ \Rightarrow e = \sqrt{1 - \frac{1}{4}}\]
\[ \Rightarrow e = \frac{\sqrt{3}}{2}\]
\[\text{ Foci } = \left( x_1 , y_1 \pm be \right)\]
\[ = \left( 1, - 1 \pm \sqrt{3} \right)\]
\[\]