Question

If an ellipse the distance between the foci is 8 and the distance between the directrices is 25. The length of the minor axis is

• A. $10\sqrt{2}$
• B. $20\sqrt{2}$
• C. $30\sqrt{2}$
• D. none of these.

Answer 1

The Correct Option is A

Let the ellipse be $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}} = 1$ Distance between foci $= 2ae = 8$ $ \Rightarrow ae = 4 $ Distance between directrices $= \frac{2a}{e} = 25$ $ \therefore \left(ar\right)\left(\frac{2a}{e}\right) = 4\times25 $ $\Rightarrow 2a^{2} = 100 $ $ \Rightarrow a^{2} = 50 $ $ \Rightarrow a= 5\sqrt{2} $ $ \Rightarrow 2a= 15\sqrt{2}$ $ \therefore$ length of major axis $= 10\sqrt{2}$