Question

The complex number $z$ satisfying the equation $\frac{Re(z)}{2+i} + \frac{Im(z)}{1+2i} = \frac{3}{1-2i}$, is

• A. 6 - 5i
• B. -4 + 5i
• C. 5 - 4i
• D. 3 - 5i
• E. 4 - 5i

Hint:

Since all denominators resulted in the same real value (5), we could skip writing the denominator early on and focus on the numerators. This is a common pattern in competitive exam questions.

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
Let \(z = x + iy\). Then \(Re(z) = x\) and \(Im(z) = y\).
We must solve the given equation by rationalizing the complex denominators.
Step 2: Key Formula or Approach:
Multiply the numerator and denominator of each term by its complex conjugate.
Equate the real and imaginary parts to form a system of equations.
Step 3: Detailed Explanation:
Substitute \(x\) and \(y\):
\[ \frac{x}{2 + i} + \frac{y}{1 + 2i} = \frac{3}{1 - 2i} \] Rationalize each term:
\[ \frac{x(2 - i)}{5} + \frac{y(1 - 2i)}{5} = \frac{3(1 + 2i)}{5} \] Multiply the entire equation by 5 to clear denominators:
\[ 2x - xi + y - 2yi = 3 + 6i \] Group real and imaginary parts:
\[ (2x + y) - i(x + 2y) = 3 + 6i \] Equate components:
1) \(2x + y = 3\)
2) \(-(x + 2y) = 6 \implies x + 2y = -6\)
Solve the system. Multiply the second equation by 2:
\[ 2x + 4y = -12 \] Subtract the first equation from it:
\[ (2x + 4y) - (2x + y) = -12 - 3 \implies 3y = -15 \implies y = -5 \] Substitute \(y = -5\) into \(2x + y = 3\):
\[ 2x - 5 = 3 \implies 2x = 8 \implies x = 4 \] Thus, \(z = 4 - 5i\).
Step 4: Final Answer:
The complex number is \(4 - 5i\).