Question:
<p>For each point (x, y) on an ellipse, the sum of the distances from (x, y) to the points (2, 0) and (-2, 0) is 8. Then the positive value of x so that (x, 3) lies on the ellipse is</p>
<p style="display:inline">4</p>
<p style="display:inline"><span class="math-tex">\(2 \sqrt{3}\)</span></p>
<p style="display:inline">2</p>
<p style="display:inline"><span class="math-tex">\(\frac{1}{\sqrt{3}}\)</span></p>
Step-by-Step Solution
Key Concept: An ellipse is defined as the locus of points where the sum of distances to two fixed foci is constant and equal to the length of the major axis (2a).
<p>2a = 8 <span class="math-tex">$\Rightarrow$</span> a = 4<br />
Let F = (2, 0) and F' = (-2, 0)<br />
<span class="math-tex">$\Rightarrow$</span> 2ae = 4 <span class="math-tex">$\Rightarrow e=\frac{1}{2}$</span><br />
b<sup>2</sup> = a<sup>2</sup>(1 - e<sup>2</sup>) = 16<span class="math-tex">$\left(1-\frac{1}{4}\right)$</span> = 12<br />
Equation of the ellipse is <span class="math-tex">$\frac{x^{2}}{16}+\frac{y^{2}}{12}=1$</span><br />
(x, 3) lies on it <span class="math-tex">$\Rightarrow$</span> x<sup>2</sup> = 4 <span class="math-tex">$\Rightarrow$</span> x = <span class="math-tex">$\pm$</span>2</p>
Correct Answer: C