Circles
Chord Bisected at a Given Point
Grade 11
Question:
<p>The middle point of the chord intercepted on the line <math>lx + my + n = 0</math> by the circle <math>x^2 + y^2 = a^2</math> is</p>
<p>(a) <math>\left(-\frac{nl}{l^2 + m^2}, -\frac{nm}{l^2 + m^2}\right)</math></p>
<p>(b) <math>\left(\frac{nl}{l^2 + m^2}, \frac{nm}{l^2 + m^2}\right)</math></p>
<p>(c) <math>\left(-\frac{l}{l^2 + m^2}, \frac{m}{l^2 + m^2}\right)</math></p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: Use the equation of chord with a given midpoint and the constraint that it lies on the given line to find the coordinates.
<p><strong>Solution:</strong> Let <math>(x_1, y_1)</math> be the middle point of the chord intercepted by the circle <math>x^2 + y^2 = a^2</math> on the line <math>lx + my + n = 0</math>.</p><p>The equation of the chord of the circle <math>x^2 + y^2 = a^2</math>, whose middle point is <math>(x_1, y_1)</math>, is:</p><p><math>xx_1 + yy_1 = x_1^2 + y_1^2</math></p><p>Since this chord lies on the line <math>lx + my + n = 0</math>, we can write:</p><p><math>\frac{x_1}{l} = \frac{y_1}{m} = \frac{a^2 - n}{l^2 + m^2}</math></p><p>Therefore, <math>x_1 = -\frac{nl}{l^2 + m^2}</math> and <math>y_1 = -\frac{nm}{l^2 + m^2}</math></p><p>∴ Answer is (a).</p>
Correct Answer: A