Parabola
Chord Properties
Grade 11
Question:
<p>The locus of the middle points of chords of the parabola \(y^2 = 4x\), which are of constant length \('2l'\) is:</p>
<p>(a) \((4x + y^2)(y^2 - 4) = 4l^2\)</p>
<p>(b) \((4y + x^2)(x^2 - 4) = 4l^2\)</p>
<p>(c) \((4y - x^2)(x^2 + 4) = 4l^2\)</p>
<p>(d) \((4x - y^2)(y^2 + 4) = 4l^2\)</p>
Step-by-Step Solution
Key Concept: Use parametric representation of parabola and chord length formula to derive relation between midpoint coordinates and constant chord length.
<p>Let \((h, k)\) be the midpoint of a chord of constant length \(2l\).</p><p>For parabola \(y^2 = 4x\), if chord has endpoints with parameters \(t_1, t_2\), then midpoint is \((t_1 t_2 + 1, t_1 + t_2)\).</p><p>Chord length: \(\sqrt{(t_1-t_2)^2(t_1^2 + t_2^2 + 2)} = 2l\)</p><p>Using relations for midpoint: \((4x - y^2)(y^2 + 4) = 4l^2\)</p>
Correct Answer: d