Complex Numbers
Geometry in Argand Plane
Grade 11
Question:
<p>If <span class='math'>z_1, z_2, z_3</span> and <span class='math'>z_4</span> are the affixes of four points in the Argand plane and <span class='math'>z</span> is the affix of a point, such that <span class='math'>|z - z_1| = |z - z_2| = |z - z_3| = |z - z_4|</span>, then <span class='math'>z_1, z_2, z_3</span> and <span class='math'>z_4</span> are</p>
<p>(a) concyclic</p>
<p>(b) collinear</p>
<p>(c) vertices of a square</p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: Equal distances from a point to four distinct points means those four points lie on a circle centered at that point.
<p><strong>Solution:</strong> The condition <span class='math'>|z - z_1| = |z - z_2| = |z - z_3| = |z - z_4|</span> means that point <span class='math'>z</span> is equidistant from all four points <span class='math'>z_1, z_2, z_3, z_4</span>.</p><p>In the Argand plane, this means <span class='math'>z</span> is the center of a circle passing through all four points. Therefore, the four points must lie on the same circle (i.e., they are concyclic).</p><p>∴ Answer is (a) concyclic.</p>
Correct Answer: A