Question:
<p>The distance between the foci of the ellipse x = 3cos<span class="math-tex">\(\theta\)</span>, y = 4sin<span class="math-tex">\(\theta\)</span> is</p>
<p style="display:inline"><span class="math-tex">\(2 \sqrt{7}\)</span></p>
<p style="display:inline"><span class="math-tex">\(7 \sqrt{2}\)</span></p>
<p style="display:inline"><span class="math-tex">\(\sqrt{7}\)</span></p>
<p style="display:inline"><span class="math-tex">\(3 \sqrt{7}\)</span></p>
Step-by-Step Solution
Key Concept: Convert the parametric equations to standard form and identify whether the major axis is horizontal or vertical to apply the correct focal distance formula.
<p>x<sup>2</sup> = 9cos<sup>2</sup><span class="math-tex">$\theta$</span>, y<sup>2</sup> = 16sin<sup>2</sup><span class="math-tex">$\theta$</span><br />
<span class="math-tex">$\Rightarrow \frac{x^{2}}{9}+\frac{y^{2}}{16}=1$</span> <span class="math-tex">$\Rightarrow$</span> a<sup>2</sup> = 9, b<sup>2</sup> = 16<br />
<span class="math-tex">$\Rightarrow e^{2}=1-\frac{a^{2}}{b^{2}} \Rightarrow e=\frac{\sqrt{7}}{4}$</span><br />
Distance between foci = 2be = <span class="math-tex">$2 \sqrt{7}$</span></p>
Correct Answer: A