Question

Find the equation of an ellipse, the distance between the foci is 8 units and the distance between the directrices is 18 units.

Answer 1

\[\text{ The distance between the foci is 8 units }.\]
\[\text{ i . e } . 2ae = 8 . . . (1)\]
\[\text{ The distance between the directrices is 18 units  }.\]
\[\text{ i . e } . \frac{2a}{e} = 18 . . . (2)\]
\[\text{ Comparing eqs. } (1) \text{ and } (2), \text{ we get }:\]
\[e=\frac{8}{2a}\]
\[\text{ Substituting the value ofein eq. } (2), \text{ we get }:\]
\[\frac{2a}{\frac{8}{2a}} = 18\]
\[ \Rightarrow 4 a^2 = 18 \times 8\]
\[ \Rightarrow a^2 = 36 or a = 6\]
\[\text{ Now }, 2ae = 8\]
\[ \Rightarrow 12e = 8 \text{ or } e = \frac{2}{3}\]
\[\text{ Then } b^2 = a^2 (1 - e^2 )\]
\[ \Rightarrow b^2 = 36\left( 1 - \frac{4}{9} \right)\]
\[ \Rightarrow b^2 = 36 \times \frac{5}{9}\]
\[ \Rightarrow b^2 = 20\]
\[ \therefore \frac{x^2}{36} + \frac{y^2}{20} = 1\]
\[\text{ This is the required equation of ellipse }.\]
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