Question

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

 25x² + 16y² = 1600.

Answer 1

\[ 25 x^2 + 16 y^2 = 1600\]
\[ \Rightarrow \frac{x^2}{64} + \frac{y^2}{100} = 1\]
\[\text{ This is of the form }  \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, \text{ where } a^2 = 64 \text{ and } b^2 = 100, i . e . a = 8 \text{ and } b = 10 . \]
\[\text{ Clearly, }  b > a\]
\[\text{ Now} , e = \sqrt{1 - \frac{a^2}{b^2}}\]
\[ \Rightarrow e = \sqrt{1 - \frac{64}{100}}\]
\[ \Rightarrow e = \sqrt{\frac{36}{100}}\]
\[ \Rightarrow e = \frac{6}{10} or \frac{3}{5}\]
\[\text{ Coordinates of the foci } = \left( 0, \pm 6 \right)\]
\[ \text{ Length of latus rectum } =\frac{2 a^2}{b}\]
\[ = \frac{2 \times 64}{10}\]
\[ = \frac{64}{5}\]