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 \( z = x + iy \). The equation involves the real part \( x \) and the imaginary part \( y \). We solve for $x$ and $y$ by rationalizing each complex fraction and comparing components.

Step 2: Detailed Explanation:
The given equation is:
\[ \frac{x}{2+i} + \frac{y}{1+2i} = \frac{3}{1-2i} \]
Rationalize each term:
\[ \frac{x(2-i)}{(2+i)(2-i)} + \frac{y(1-2i)}{(1+2i)(1-2i)} = \frac{3(1+2i)}{(1-2i)(1+2i)} \]
\[ \frac{x(2-i)}{5} + \frac{y(1-2i)}{5} = \frac{3(1+2i)}{5} \]
Multiply the entire equation by 5:
\[ 2x - xi + y - 2yi = 3 + 6i \]
Separate real and imaginary parts on the LHS:
\[ (2x + y) + i(-x - 2y) = 3 + 6i \]
Comparing parts:
1) \( 2x + y = 3 \)
2) \( -x - 2y = 6 \implies x + 2y = -6 \)
From (1), \( y = 3 - 2x \). Substitute into (2):
\[ x + 2(3 - 2x) = -6 \]
\[ x + 6 - 4x = -6 \]
\[ -3x = -12 \implies x = 4 \]
Substitute $x=4$ into \( y = 3 - 2x \):
\[ y = 3 - 2(4) = -5 \]
So, \( z = 4 - 5i \).

Step 3: Final Answer:
The complex number is $4 - 5i$.