Question

If the conjugate of \( (x + iy)(1 - 2i) \) be \( 1 + i \), then:

• A. \( x = \frac{1}{5} \)
• B. \( y = \frac{3}{5} \)
• C. \( x - iy = \frac{1 - i}{1 + 2i} \)
• D. \( x + iy = \frac{1 - i}{1 - 2i} \)

Hint:

When dealing with conjugates of complex numbers, separate the real and imaginary parts and use the given conditions to solve the system of equations.

Answer 1

The Correct Option is D

We are given that the conjugate of \( (x + iy)(1 - 2i) \) is \( 1 + i \). First, express the product \( (x + iy)(1 - 2i) \): \[ (x + iy)(1 - 2i) = x(1 - 2i) + iy(1 - 2i) \] Expanding the terms: \[ x(1 - 2i) = x - 2ix \] \[ iy(1 - 2i) = iy - 2i^2y = iy + 2y \] Thus: \[ (x + iy)(1 - 2i) = (x + 2y) + i(y - 2x) \] The conjugate of this expression is: \[ (x + 2y) - i(y - 2x) \] We are given that the conjugate is \( 1 + i \), so we equate real and imaginary parts: \[ x + 2y = 1 \quad \text{(real part)} \] \[ y - 2x = 1 \quad \text{(imaginary part)} \] Solving this system of equations: 1. From \( x + 2y = 1 \), we get \( x = 1 - 2y \). 2. Substitute \( x = 1 - 2y \) into \( y - 2x = 1 \): \[ y - 2(1 - 2y) = 1 \] \[ y - 2 + 4y = 1 \] \[ 5y = 3 \] \[ y = \frac{3}{5} \] Substitute \( y = \frac{3}{5} \) into \( x = 1 - 2y \): \[ x = 1 - 2 \times \frac{3}{5} = 1 - \frac{6}{5} = \frac{-1}{5} \] Thus, the values of \( x \) and \( y \) are \( x = \frac{-1}{5} \) and \( y = \frac{3}{5} \). Finally, the expression for \( x + iy \) is: \[ x + iy = \frac{1 - i}{1 - 2i} \] Thus, the correct answer is option (D)