Complex Numbers
Locus of Complex Numbers
Grade None
Question:
<p>If <span class="math">\(\omega = \frac{z}{z - i}\)</span> and <span class="math">\(|\omega| = 1\)</span>, where <span class="math">\(i = \sqrt{-1}\)</span>, then <span class="math">z</span> lies on</p>
<p>(a) a straight line</p>
<p>(b) a parabola</p>
<p>(c) an ellipse</p>
<p>(d) a circle</p>
Step-by-Step Solution
Key Concept: The condition |z| = |z - i| represents all points equidistant from two fixed points, forming a straight line.
<p><strong>Solution:</strong> Given <span class="math">$\omega = \frac{z}{z - i}$</span> and <span class="math">$|\omega| = 1$</span>.</p><p>From <span class="math">$|\omega| = 1$</span>, we have <span class="math">$\left|\frac{z}{z - i}\right| = 1$</span>.</p><p>This gives <span class="math">$|z| = |z - i|$</span>.</p><p>This is the locus of points equidistant from the origin and the point <span class="math">i$</span>, which is a straight line (the perpendicular bisector of the segment joining <span class="math">0$</span> and <span class="math">i$</span>).</p><p>∴ Answer is (a).</p>
Correct Answer: A