Question

Find the equation of an ellipse whose vertices are (0, ± 10) and eccentricity e = \[\frac{4}{5}\] 

Answer 1

\[\text{ Let the equation of the required ellipse be }\]
\[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 . . . (1)\]
\[\text{ Since the vertices of the ellipse are on the y - axis, the coordinates of the vertices are } (0, \pm b) . \]
\[ \therefore b = 10\]
\[\text{ Now }, a^2 = b^2 (1 - e^2 )\]
\[ \Rightarrow a^2 = 100 \left( 1 - \frac{4}{5} \right)^2 \]
\[ \Rightarrow a^2 = 100 \times \left( \frac{9}{25} \right)\]
\[ \Rightarrow a^2 = 36\]
\[\text{ Substituting the values of a^2 and b^2 in equation (1), we get }: \]
\[\frac{x^2}{36} + \frac{y^2}{100} = 1\]
\[ \Rightarrow \frac{100 x^2 + 36 y^2}{3600} = 1\]
\[ \therefore 100 x^2 + 36 y^2 = 3600\]
\[\text{ This is the required equation of the ellipse } . \]
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