Complex Numbers
Locus of Complex Numbers
Grade 11
Question:
<p>The locus of the points <span class='math'>z</span> which satisfy the condition <span class='math'>\arg\left(\frac{z-1}{z+1}\right) = \frac{\pi}{3}</span> is</p>
<p>(a) a straight line</p>
<p>(b) a circle</p>
<p>(c) a parabola</p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: The locus of points from which a fixed line segment subtends a constant angle is a circular arc.
<p><strong>Solution:</strong> The condition <span class='math'>\arg\left(\frac{z-1}{z+1}\right) = \frac{\pi}{3}</span> means the angle at point <span class='math'>z</span> subtended by the line segment joining <span class='math'>-1</span> and <span class='math'>1</span> is constant at <span class='math'>\frac{\pi}{3}</span>.</p><p>By the locus theorem, the locus of points from which a line segment subtends a constant angle is a circular arc (part of a circle passing through the endpoints of the segment).</p><p>∴ Answer is (b) a circle.</p>
Correct Answer: B