Question

In the Argand plane, the values of \( z \) satisfying the equation \( |z - 1| = |i(z + 1)| \) lie on:

• A. the Y-axis
• B. a Parabola
• C. a Hyperbola
• D. the X-axis

Hint:

When solving modulus equations involving complex numbers, convert \( z = x + iy \) and compare magnitudes to derive geometric loci.

Answer 1

The Correct Option is A

Let \( z = x + iy \). Then, \[ |z - 1| = \sqrt{(x - 1)^2 + y^2}, \quad |i(z + 1)| = |i(x + 1 + iy)| = \sqrt{(x + 1)^2 + y^2} \] Equating: \[ (x - 1)^2 + y^2 = (x + 1)^2 + y^2 \Rightarrow x^2 - 2x + 1 = x^2 + 2x + 1 \Rightarrow -2x = 2x \Rightarrow x = 0 \] So, the locus is \( x = 0 \), i.e., the Y-axis.