Complex Numbers
Modulus and Argument
Grade Class 11
Question:
<p>If \( |z - z_1| = |z - z_2| \) and \( \arg\left(\dfrac{z - z_1}{z + z_1}\right) = \dfrac{\pi}{2} \), then \( z \) lies on:</p>
Midpoint of z_1z_2
Perpendicular bisector of z_1z_2
Circle with z_1z_2 as diameter
Line through z_1 and z_2
Step-by-Step Solution
Key Concept: arg((z-z_1)/(z-(-z_1))) = \pi/2 means angle subtended at z by the segment from z_1 to -z_1 is \pi/2, so z lies on a circle with diameter from z_1 to -z_1.
<p>The condition \( \arg\left(\dfrac{z-z_1}{z+z_1}\right) = \dfrac{\pi}{2} \) means the angle at \(z\) in the triangle formed with \(z_1\) and \(-z_1\) is \(\pi/2\). So \(z\) lies on the circle with diameter \(z_1(-z_1)\).</p>
Correct Answer: C