Question:
<p>If the length of the minor axis of ellipse is equal to half of the distance between the foci, then the eccentricity of the ellipse is:</p>
<p style="display:inline"><span class="math-tex">\(\frac{\sqrt{3}}{2}\)</span></p>
<p style="display:inline"><span class="math-tex">\(\frac{\sqrt{5}}{3}\)</span></p>
<p style="display:inline"><span class="math-tex">\(\frac{1}{\sqrt{3}}\)</span></p>
<p style="display:inline"><span class="math-tex">\(\frac{2}{\sqrt{5}}\)</span></p>
Step-by-Step Solution
Key Concept: Use the fundamental relationship $b^2 = a^2(1-e^2)$ to solve for eccentricity by substituting the given ratio between the minor axis ($2b$) and the distance between foci ($2ae$).
<p>According to question, <span class="math-tex">\(2 b=a e\)</span><br />
<span class="math-tex">\(\Rightarrow \frac{b}{a}=\frac{e}{2}\)</span><br />
<span class="math-tex">\(e=\sqrt{1-\frac{e^{2}}{4}}\)</span><br />
Hence <span class="math-tex">\(e=\frac{2}{\sqrt{5}}\)</span></p>
Correct Answer: D