Question:
<p>If the distance between the directrices be thrice the distance between the foci, then the eccentricity of an ellipse is:</p>
<p style="display:inline"><span class="math-tex">\(\frac 23\)</span></p>
<p style="display:inline"><span class="math-tex">\(\frac 12\)</span></p>
<p style="display:inline"><span class="math-tex">\(\frac 45\)</span></p>
<p style="display:inline"><span class="math-tex">\(\frac 1{\sqrt 3}\)</span></p>
Step-by-Step Solution
Key Concept: Use the standard formulas for the distance between directrices (2a/e) and the distance between foci (2ae) to establish an equation and solve for the eccentricity.
<p>According to the condition, <span class="math-tex">$\frac{2 a}{e}$</span> = 6 ae<br />
<span class="math-tex">$\Rightarrow$</span> e = <span class="math-tex">$\frac 1{\sqrt 3}$</span></p>
Correct Answer: D