Question:
<p>If the length of the major axis of an ellipse is three times the length of its minor axis, then its eccentricity is</p>
<p style="display:inline"><span class="math-tex">\(\frac{1}{\sqrt{2}}\)</span></p>
<p style="display:inline"><span class="math-tex">\(\frac{1}{\sqrt{3}}\)</span></p>
<p style="display:inline"><span class="math-tex">\(\frac{1}{3}\)</span></p>
<p style="display:inline"><span class="math-tex">\(\frac{2 \sqrt{2}}{3}\)</span></p>
Step-by-Step Solution
Key Concept: The eccentricity of an ellipse is calculated by substituting the given ratio of the major and minor axes into the fundamental relationship $b^2 = a^2(1 - e^2)$.
<p>Given, Major axis = 3(Minor axis) <span class="math-tex">$\Leftrightarrow$</span> 2a = 3(2b)<br />
<span class="math-tex">$\Leftrightarrow$</span> a = 3b <span class="math-tex">$\Rightarrow$</span> a<sup>2</sup> = 9b<sup>2</sup> = 9a<sup>2</sup>(1 - e<sup>2</sup>)<br />
<span class="math-tex">$\Rightarrow e=\frac{2 \sqrt{2}}{3}$</span></p>
Correct Answer: D