Question

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

x² + 4y² − 2x = 0 

Answer 1

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