Question

Find the locus of \( Z = x + iy \) satisfying \[ \frac{\text{Re}(z)}{2 + i} + \frac{\text{Im}(z)}{1 + 2i} = \frac{3}{1 - 2i} \]

• A. \( x^2 + y^2 = 1 \)
• B. \( x^2 + y^2 = 9 \)
• C. \( x^2 + y^2 = 4 \)
• D. \( x^2 + y^2 = 25 \)

Hint:

When solving for the locus of a complex number, express real and imaginary parts separately, then equate the real and imaginary components.

Answer 1

The Correct Option is C

Step 1: Express real and imaginary parts.
We are given: \[ \frac{\text{Re}(z)}{2 + i} + \frac{\text{Im}(z)}{1 + 2i} = \frac{3}{1 - 2i} \] Let \( z = x + iy \), where \( x = \text{Re}(z) \) and \( y = \text{Im}(z) \).
Step 2: Simplify the equation.
First, simplify \( \frac{3}{1 - 2i} \) by multiplying the numerator and denominator by the complex conjugate of the denominator \( 1 + 2i \): \[ \frac{3}{1 - 2i} \times \frac{1 + 2i}{1 + 2i} = \frac{3(1 + 2i)}{(1 - 2i)(1 + 2i)} = \frac{3(1 + 2i)}{1^2 + (2)^2} = \frac{3(1 + 2i)}{5} = \frac{3}{5} + \frac{6i}{5} \] Now, simplify the left-hand side: \[ \frac{x}{2 + i} + \frac{y}{1 + 2i} \] Multiply the first term by \( \frac{2 - i}{2 - i} \) and the second term by \( \frac{1 - 2i}{1 - 2i} \) to rationalize: \[ \frac{x(2 - i)}{(2 + i)(2 - i)} + \frac{y(1 - 2i)}{(1 + 2i)(1 - 2i)} \] \[ = \frac{x(2 - i)}{5} + \frac{y(1 - 2i)}{5} \] This simplifies to: \[ \frac{2x - ix + y - 2iy}{5} = \frac{3}{5} + \frac{6i}{5} \]
Step 3: Equate real and imaginary parts.
Equating real and imaginary parts gives the system of equations: \[ \frac{2x + y}{5} = \frac{3}{5} \quad \Rightarrow \quad 2x + y = 3 \] \[ \frac{-x - 2y}{5} = \frac{6}{5} \quad \Rightarrow \quad -x - 2y = 6 \] Solving these equations, we find: \[ 2x + y = 3 \] \[ -x - 2y = 6 \] Multiplying the second equation by 2: \[ -2x - 4y = 12 \] Now adding the equations: \[ 2x + y + (-2x - 4y) = 3 + 12 \] \[ -3y = 15 \quad \Rightarrow \quad y = -5 \] Substitute \( y = -5 \) into \( 2x + y = 3 \): \[ 2x - 5 = 3 \quad \Rightarrow \quad 2x = 8 \quad \Rightarrow \quad x = 4 \] Thus, the locus of \( Z \) is \( x^2 + y^2 = 4 \).
Final Answer: \( x^2 + y^2 = 4 \).