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प्रश्न
Find the equation of the standard ellipse, taking its axes as the coordinate axes, whose minor axis is equal to the distance between the foci and whose length of the latus rectum is 10. Also, find its eccentricity.
उत्तर
\[\text{ According to the question, the minor axis is equal to the distance between the foci }.\]
\[\text{ i . e } . 2b = 2\text{ ae and } \frac{{2b}^2}{a} = 10 \text{ or } b^2 = 5a \]
\[ \Rightarrow b = ae\]
\[ \Rightarrow b^2 = a^2 e^2 \]
\[ \Rightarrow b^2 = a^2 \left( 1 - \frac{b^2}{a^2} \right) \left( \because e = \sqrt{1 - \frac{b^2}{a^2}} \right)\]
\[ \Rightarrow b^2 = a^2 - b^2 \]
\[ \Rightarrow a^2 = 2 b^2 \]
\[ \Rightarrow a^2 = 10a \left( \because b^2 = 5a \right)\]
\[ \Rightarrow a = 10\]
\[ \Rightarrow b^2 = 5a \]
\[ \Rightarrow b^2 = 50\]
\[\text{ Substituting the values of a and b in the equation of an ellipse, we get }:\]
\[\frac{x^2}{100} + \frac{y^2}{50} = 1\]
\[ \therefore x^2 + 2 y^2 = 100\]
\[\text{This is the required equation of the ellipse }.\]
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