Question:
<p>The equation of the ellipse whose centre is at origin and which passes through the points (-3, 1) and (2, -2) is</p>
<p style="display:inline">3x<sup>2</sup> + 5y<sup>2</sup> = 32</p>
<p style="display:inline">5x<sup>2</sup> - 3y<sup>2</sup> = 32</p>
<p style="display:inline">5x<sup>2</sup> + 3y<sup>2</sup> = 32</p>
<p style="display:inline">3x<sup>2</sup> + 5y<sup>2</sup> + 32 = 0</p>
Step-by-Step Solution
Key Concept: Substitute the coordinates of the given points into the standard ellipse equation x²/a² + y²/b² = 1 to form and solve a system of linear equations for 1/a² and 1/b².
<p>Let the equation of the ellipse be <span class="math-tex">$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$</span>.<br />
Since it passes through (-3, 1) and (2, -2)<br />
<span class="math-tex">$\therefore \frac{9}{a^{2}}+\frac{1}{b^{2}}=1$</span> and <span class="math-tex">$\frac{1}{a^{2}}+\frac{1}{b^{2}}=\frac{1}{4}$</span><br />
By solving both equations, we get<br />
<span class="math-tex">$a^{2}=\frac{32}{3}, b^{2}=\frac{32}{5}$</span><br />
Hence, the required equation of the ellipse is 3x<sup>2</sup> + 5y<sup>2</sup> = 32.</p>
Correct Answer: A