Question

If a complex number $ z = x + iy $ represents a point $ P $ on the Argand plane and $$ \text{Arg} \left( \frac{z - 3 + 2i}{z + 2 - 3i} \right) = \frac{\pi}{4} $$ then the locus of $ P $ is

• A. Circle with the line \( x + y = 12 \) as its diameter
• B. Circle with radius \( \sqrt{11} \)
• C. Circle with the line \( x - y = 6 \) as its diameter
• D. Circle with radius 5

Hint:

An argument equation like \( \text{Arg} \left( \frac{z - z_1}{z - z_2} \right) = \theta \) represents a circle when \( \theta \) is constant.

Answer 1

The Correct Option is D

Let \( z = x + iy \). Then, \[ \text{Arg} \left( \frac{z - (3 - 2i)}{z - (-2 + 3i)} \right) = \frac{\pi}{4} \Rightarrow \angle APB = \frac{\pi}{4} \] This represents all points \( P \) such that angle \( APB = \frac{\pi}{4} \), i.e., \( P \) lies on a circle subtending a constant angle at the points \( A \) and \( B \). The geometric locus is a circle. Calculating the distance and geometry, radius turns out to be 5.