Question

If \( z + \bar{z} = 6 \) and \( z - \bar{z} = 4i \), then \( |z|^2 = \)

• A. \(36 \)
• B. \(16 \)
• C. \(15 \)
• D. \(13 \)
• E. \(9 \)

Hint:

Use \( z + \bar{z} = 2x \) and \( z - \bar{z} = 2iy \) to quickly extract real and imaginary parts.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
This problem uses the properties of a complex number `z` and its conjugate `\bar{z}` to find its components and then calculate the square of its modulus.
Step 2: Key Formula or Approach:
Let the complex number be `z = x + iy`, where `x` is the real part (Re(z)) and `y` is the imaginary part (Im(z)).
Its conjugate is `\bar{z} = x - iy`.
The following identities are key:
1. `z + \bar{z} = (x + iy) + (x - iy) = 2x = 2Re(z)`
2. `z - \bar{z} = (x + iy) - (x - iy) = 2iy = 2i \cdot Im(z)`
The square of the modulus is `|z|^2 = x^2 + y^2`.
Step 3: Detailed Explanation:
We are given two equations:
1. `z + \bar{z} = 6`
Using the identity `z + \bar{z} = 2x`, we have:
\[ 2x = 6 \implies x = 3 \] 2. `z - \bar{z} = 4i`
Using the identity `z - \bar{z} = 2iy`, we have:
\[ 2iy = 4i \implies 2y = 4 \implies y = 2 \] So, the complex number is `z = x + iy = 3 + 2i`.
Now, we need to find `|z|^2`.
\[ |z|^2 = x^2 + y^2 = 3^2 + 2^2 \] \[ |z|^2 = 9 + 4 = 13 \] Step 4: Final Answer:
The value of |z|\(^2\) is 13.