Question

Find the equation of the ellipse in the following case: 

Length of major axis 26, foci (± 5, 0) 

Answer 1

\[\text{ Length of major axis }=26\]
\[\text{ Foci }=\left( \pm 5, 0 \right)\]
\[\text{ We have } 2a = 26\]
\[ \Rightarrow a = 13\]
\[\text{ Also }, ae = 5\]
\[ \Rightarrow e = \frac{5}{13}\]
\[\text{ Now }, e = \sqrt{1 - \frac{b^2}{a^2}}\]
\[ \Rightarrow \frac{5}{13} = \sqrt{1 - \frac{b^2}{169}}\]
\[\text{ On squaring both sides, we get }:\]
\[\frac{25}{169} = \frac{169 - b^2}{169}\]
\[ \Rightarrow b^2 = 144\]
\[\text{ Now }, \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\]
\[ \Rightarrow \frac{x^2}{169} + \frac{y^2}{144} = 1\]
\[\text{ This is the required equation of the ellipse }.\]