Question

If the amplitude of \( (Z - 2) \) is \( \frac{\pi}{2} \), then the locus of \( Z \) is:

• A. \( x = 0, \, y>0 \)
• B. \( x = 2, \, y>0 \)
• C. \( x>0, \, y = 2 \)
• D. \( x>0, \, y = 0 \)

Hint:

The argument of a complex number \( Z = x + iy \) represents the angle that the point \( (x, y) \) makes with the real axis. Use the condition \( \arg(Z) = \frac{\pi}{2} \) to identify points along the imaginary axis.

Answer 1

The Correct Option is B

We are given that the amplitude of \( (Z - 2) \) is \( \frac{\pi}{2} \). The amplitude (or argument) of a complex number \( Z - 2 \) is the angle \( \theta \) that the vector representing \( Z - 2 \) makes with the positive real axis. This means: \[ \arg(Z - 2) = \frac{\pi}{2}. \] This implies that the line joining the origin to the point \( Z - 2 \) makes an angle of \( \frac{\pi}{2} \) with the real axis, which means it lies along the imaginary axis. Therefore, the real part of \( Z \) is constant and equal to 2. 
Step 1: If \( Z = x + iy \), then \( Z - 2 = (x - 2) + iy \). The argument of \( Z - 2 \) is given by: \[ \arg(Z - 2) = \arg((x - 2) + iy). \] Since the argument is \( \frac{\pi}{2} \), this means that \( x - 2 = 0 \), implying that \( x = 2 \). 
Step 2: Thus, the locus of \( Z \) is a vertical line at \( x = 2 \), with \( y \) being any real number. Therefore, the condition for the locus of \( Z \) is \( x = 2, \, y>0 \).