Question

Let \[ S=\{z: z^2+4z+16=0,\; z\in\mathbb{C}\} \] then the value of \[ \sum_{z\in S}|z+\sqrt{3}i|^2 \] is

• A. \(34\)
• B. \(35\)
• C. \(38\)
• D. \(41\)

Hint:

For a complex number \(a+bi\), \(|a+bi|^2=a^2+b^2\).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
First, solve the quadratic equation to find the complex roots \( z_1 \) and \( z_2 \). Then substitute them into the absolute value expression.
Step 2: Key Formula or Approach:
For \( z = x + iy \), \( |z|^2 = x^2 + y^2 \).
Step 3: Detailed Explanation:
1. Solve \( z^2 + 4z + 16 = 0 \): \[ z = \frac{-4 \pm \sqrt{16 - 64}}{2} = \frac{-4 \pm \sqrt{-48}}{2} = -2 \pm 2\sqrt{3}i \] So, \( z_1 = -2 + 2\sqrt{3}i \) and \( z_2 = -2 - 2\sqrt{3}i \). 2. Calculate first term: \( |z_1 + \sqrt{3}i|^2 = |-2 + 2\sqrt{3}i + \sqrt{3}i|^2 = |-2 + 3\sqrt{3}i|^2 \) \[ = (-2)^2 + (3\sqrt{3})^2 = 4 + 27 = 31 \] 3. Calculate second term: \( |z_2 + \sqrt{3}i|^2 = |-2 - 2\sqrt{3}i + \sqrt{3}i|^2 = |-2 - \sqrt{3}i|^2 \) \[ = (-2)^2 + (-\sqrt{3})^2 = 4 + 3 = 7 \] 4. Sum: \( 31 + 7 = 38 \).
Step 4: Final Answer:
The value of the sum is 38.