Question

If the distance between the directrices of a rectangular hyperbola are $10$, then the distance between its foci will be

• A. $10\sqrt{2}$
• B. $5$
• C. $5\sqrt{2}$
• D. $20$

Answer 1

The correct option is D

$20$

Explanation for correct option

Step 1: Formula used

The equation of the hyperbola is $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$

1. Distance between foci is $=2ae$
2. Distance between the directrices $=\frac{2a}{e}$
3. The eccentricity of the hyperbola $e=\sqrt{1+\frac{b^2}{a^2}}$

Step 2: Solve for the distance between the foci

We know that the equation of rectangular hyperbola is $x^2-y^2=a^2$

Therefore eccentricity $e=\sqrt{1+\frac{b^2}{a^2}}=\sqrt{2}\left[\because b=a\right]$

Given the distance between the directrices of a rectangular hyperbola are $10$,

$$\begin{gathered} \therefore \frac{2a}{e}=10 \\ \Rightarrow \frac{2a}{\sqrt{2}}=10 \\ \Rightarrow a=5\sqrt{2} \end{gathered}$$

Therefore the distance between the foci $=2ae=2·5\sqrt{2}.\sqrt{2}=20$

Hence, option (D) is correct