Question:
<p>The latus rectum of an ellipse is 10 and the minor axis is equal to the distance between the foci. The equation of the ellipse is:</p>
<p style="display:inline">x<sup>2</sup> - 2y<sup>2</sup> = 100</p>
<p style="display:inline">x<sup>2</sup> - <span class="math-tex">\(\sqrt 2\)</span>y<sup>2</sup> = 10</p>
<p style="display:inline">x<sup>2</sup> + <span class="math-tex">\(\sqrt 2\)</span>y<sup>2</sup> = 10</p>
<p style="display:inline">x<sup>2</sup> + 2y<sup>2</sup> = 100</p>
Step-by-Step Solution
Key Concept: Solve for the ellipse parameters by applying the latus rectum formula 2b²/a and the relationship b² = a²(1 - e²) based on the given geometric conditions.
<p>Given <span class="math-tex">$\frac{2 b^{2}}{a}$</span> = 10 and 2b = 2ae<br />
Also b<sup>2</sup> = a<sup>2</sup>(1 - e<sup>2</sup>)<br />
<span class="math-tex">$\Rightarrow$</span> e<sup>2</sup> = (1 - e<sup>2</sup>)<br />
<span class="math-tex">$\Rightarrow$</span> e = <span class="math-tex">$\frac{1}{\sqrt{2}}$</span><br />
<span class="math-tex">$\Rightarrow$</span> b = <span class="math-tex">$\frac{a}{\sqrt{2}}$</span> or b = 5<span class="math-tex">$\sqrt 2$</span>, a = 10<br />
Hence equation of ellipse is <span class="math-tex">$\frac{x^{2}}{(10)^{2}}+\frac{y^{2}}{(5 \sqrt{2})^{2}}$</span> = 1<br />
i.e., x<sup>2</sup> + 2y<sup>2</sup> = 100</p>
Correct Answer: D