Question:
<p>If the foci of an ellipse are (0, <span class="math-tex">\(\pm\)</span>4) and the equation of the directrices are y = <span class="math-tex">\(\pm\)</span>9, then the equation of the ellipse is</p>
<p style="display:inline">5x<sup>2</sup> + 3y<sup>2</sup> = 45</p>
<p style="display:inline">10x<sup>2</sup> + 9y<sup>2</sup> = 180</p>
<p style="display:inline">5x<sup>2</sup> + 3y<sup>2</sup> = 60</p>
<p style="display:inline">9x<sup>2</sup> + 5y<sup>2</sup> = 180</p>
Step-by-Step Solution
Key Concept: Identify the ellipse's vertical orientation from its y-axis foci to correctly relate the parameters be and b/e to the given coordinates.
<p>Foci of an ellipse are (0, <span class="math-tex">$\pm$</span>4).<br />
<span class="math-tex">$\Rightarrow$</span> Distance between foci = 8 <span class="math-tex">$\Leftrightarrow$</span> 2be = 8<br />
<span class="math-tex">$\Rightarrow$</span> be = 4 ...(i)<br />
Also, equation of directrices are<br />
y = <span class="math-tex">$\pm$</span>9 <span class="math-tex">$\Leftrightarrow \frac{b}{e}$</span> = 9 ...(ii)<br />
Solving (i) and (ii), we get<br />
b = 6 and <span class="math-tex">$e=\frac{2}{3}$</span><br />
a<sup>2</sup> = b<sup>2</sup>(1 - e<sup>2</sup>) = <span class="math-tex">$36\left(1-\frac{4}{9}\right)$</span> = 20<br />
<span class="math-tex">$\Rightarrow$</span> Required equation of the ellipse is<br />
<span class="math-tex">$\frac{x^{2}}{20}+\frac{y^{2}}{36}=1$</span> <span class="math-tex">$\Leftrightarrow$</span> 9x<sup>2</sup> + 5y<sup>2</sup> = 180</p>
Correct Answer: D