Question:
<p>The foci of the ellipse 25x<sup>2</sup> + 50x + 9y<sup>2</sup> + 36y - 164 = 0 are</p>
<p style="display:inline">(-1, 2) and (-1, -6)</p>
<p style="display:inline">(1, -2) and (1, -6)</p>
<p style="display:inline">(-1, -2) and (1, 6)</p>
<p style="display:inline">(-1, 2) and (6, 1)</p>
Step-by-Step Solution
Key Concept: Transform the general quadratic equation into standard form by completing the square to identify the center and major axis orientation.
<p>We have, <span class="math-tex">$\frac{(x+1)^{2}}{\frac{225}{25}}+\frac{(y+2)^{2}}{\frac{225}{9}}=1$</span><br />
Here, <span class="math-tex">$a=\sqrt{\frac{225}{25}}=3, b=\sqrt{\frac{225}{9}}=5, e^{2}=1-\frac{a^{2}}{b^{2}}$</span><br />
<span class="math-tex">$\Rightarrow e=\sqrt{1-\frac{9}{25}}=\frac{4}{5}$</span><br />
Focus = (-1, -2 <span class="math-tex">$\pm$</span> 4)<br />
= (-1, 2) and (-1, -6)</p>
Correct Answer: A