<p>The locus of the middle points of chords of an ellipse <\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1>, which passes through a fixed point, is</p>
Step-by-Step Solution
Key Concept: The locus of midpoints of chords passing through a fixed point on an ellipse forms another ellipse. Use the chord equation in terms of its midpoint coordinates.
<p><strong>Solution:</strong></p><p>Let <\P(x_1, y_1)> be the middle point of chord AB of the ellipse <\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1>.</p><p>The equation of chord AB using the middle point chord formula is:</p><p><\T = S_1></p><p><\frac{xx_1}{a^2} + \frac{yy_1}{b^2} - 1 = \frac{x_1^2}{a^2} + \frac{y_1^2}{b^2} - 1></p><p><\frac{xx_1}{a^2} + \frac{yy_1}{b^2} = \frac{x_1^2}{a^2} + \frac{y_1^2}{b^2} \quad \ldots(i)></p><p>If this chord passes through a fixed point <\Q(h, k)>, then <\h> and <\k> must satisfy equation (i):</p><p><\frac{hx_1}{a^2} + \frac{ky_1}{b^2} = \frac{x_1^2}{a^2} + \frac{y_1^2}{b^2}></p><p>This is the locus equation of the middle points <$x_1, y_1)>, which represents <strong>an ellipse</strong>.</p>
Correct Answer: A