Question:
<p>The distance between the directrices of a rectangular hyperbola is 10 units, then distance between its foci is</p>
<p style="display:inline"><span class="math-tex">\(10 \sqrt{2}\)</span></p>
<p style="display:inline">5</p>
<p style="display:inline">20</p>
<p style="display:inline"><span class="math-tex">\(5 \sqrt{2}\)</span></p>
Step-by-Step Solution
Key Concept: For a rectangular hyperbola, the eccentricity is always √2, allowing the distance between foci (2ae) to be expressed as e² times the distance between directrices (2a/e).
<p>Since, distance between directrices = <span class="math-tex">$\frac{2 a}{e}$</span>and eccentricity of rectangular hyperbola = <span class="math-tex">$\sqrt{2}$</span>.<br />
The distance between directrices <span class="math-tex">$=\frac{2 a}{\sqrt{2}}$</span><br />
Given that, <span class="math-tex">$\frac{2 a}{\sqrt{2}}$</span> = 10<br />
<span class="math-tex">$\Rightarrow$</span> 2a = <span class="math-tex">$10 \sqrt{2}$</span><br />
The distance between foci = 2ae<br />
<span class="math-tex">$=(10 \sqrt{2})(\sqrt{2})$</span><br />
= 20</p>
Correct Answer: C