Question

Find the eccentricity, coordinates of foci, length of the latus-rectum of the ellipse:

 9x² + 25y² = 225

 

Answer 1

\[9 x^2 + 25 y^2 = 225\]
\[ \Rightarrow \frac{x^2}{25} + \frac{y^2}{9} = 1\]
\[\text{ This is of the form } \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, \text{ where } a^2 = 25 \text{ and } b^2 = 9, i . e . a = 5 \text{ and } b = 3 . \]
\[\text{ Clearly, } a > b\]
\[\text{ Now, } e = \sqrt{1 - \frac{b^2}{a^2}}\]
\[ \Rightarrow e = \sqrt{1 - \frac{9}{25}}\]
\[ \Rightarrow e = \sqrt{\frac{16}{25}}\]
\[ \Rightarrow e = \frac{4}{5} \]
\[\text{ Coordinates of the foci }  = \left( \pm ae, 0 \right) = \left( \pm 4, 0 \right)\]
\[\text{ Length of latus rectum } =\frac{2 b^2}{a}\]
\[ = \frac{2 \times 9}{5}\]
\[ = \frac{18}{5}\]