Question

The solution of the equation \( |z| = z + 1 + 2i \) is:

• A. \( \frac{3}{2} - 2i \)
• B. \( 3 - i \)
• C. \( 3+2i \)
• D. none of these

Hint:

When solving equations involving complex numbers, letting \( z = x + iy \) and equating the real and imaginary parts is a common strategy. Remember that \( |z| \) is always a non-negative real number. When squaring both sides of an equation like \( \sqrt{A} = B \), ensure that \( B \ge 0 \).

Answer 1

The Correct Option is A

Step 1: Let the complex number \( z \) be represented as \( z = x + iy \), where \( x \) and \( y \) are real numbers. Step 2: Substitute \( z = x + iy \) into the given equation \( |z| = z + 1 + 2i \). Recall that \( |z| = |x + iy| = \sqrt{x^2 + y^2} \). The equation becomes: \[ \sqrt{x^2 + y^2} = (x + iy) + 1 + 2i \] Step 3: Group the real and imaginary parts on the right side of the equation. \[ \sqrt{x^2 + y^2} = (x + 1) + i(y + 2) \] Step 4: Equate the real and imaginary parts on both sides. The left side, \( \sqrt{x^2 + y^2} \), is a real number. Therefore, its imaginary part is 0. Equating the imaginary parts: \[ 0 = y + 2 \quad \Rightarrow \quad y = -2 \] Equating the real parts: \[ \sqrt{x^2 + y^2} = x + 1 \] Step 5: Substitute the value \( y = -2 \) into the equation for the real parts. \[ \sqrt{x^2 + (-2)^2} = x + 1 \] \[ \sqrt{x^2 + 4} = x + 1 \] Step 6: Solve the equation for \( x \). Before squaring both sides, note that \( \sqrt{x^2 + 4} \ge 0 \), so we must have \( x + 1 \ge 0 \), which means \( x \ge -1 \). Square both sides: \[ (\sqrt{x^2 + 4})^2 = (x + 1)^2 \] \[ x^2 + 4 = x^2 + 2x + 1 \] Subtract \( x^2 \) from both sides: \[ 4 = 2x + 1 \] \[ 3 = 2x \] \[ x = \frac{3}{2} \] Step 7: Verify the condition \( x \ge -1 \). Since \( x = \frac{3}{2} \), and \( \frac{3}{2} > -1 \), the value of \( x \) is valid. Step 8: Write the solution for \( z \) and compare with options. The solution is \( z = x + iy = \frac{3}{2} + i(-2) = \frac{3}{2} - 2i \). This matches option (A).