Question

Let \(S = \{z \in \mathbb{C} : z^2 + 4z + 16 = 0\}\). Then \(\sum_{z \in S} |z + \sqrt{3}i|^2\) is equal to:

• A. 42
• B. 23
• C. 27
• D. 38

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
We need to find the roots of the quadratic equation in complex numbers and then calculate the sum of the squared distances from a specific point \(-\sqrt{3}i\) on the complex plane.

Step 2: Key Formula or Approach:
Roots of \(az^2 + bz + c = 0\) are \(z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
Property: \(|x + iy|^2 = x^2 + y^2\).

Step 3: Detailed Explanation:
The equation is \(z^2 + 4z + 16 = 0\).
\(z = \frac{-4 \pm \sqrt{16 - 64}}{2} = \frac{-4 \pm \sqrt{-48}}{2} = \frac{-4 \pm 4\sqrt{3}i}{2}\).
Roots are \(z_1 = -2 + 2\sqrt{3}i\) and \(z_2 = -2 - 2\sqrt{3}i\).
Sum \(= |z_1 + \sqrt{3}i|^2 + |z_2 + \sqrt{3}i|^2\).
Term 1: \(|-2 + 2\sqrt{3}i + \sqrt{3}i|^2 = |-2 + 3\sqrt{3}i|^2 = (-2)^2 + (3\sqrt{3})^2 = 4 + 27 = 31\).
Term 2: \(|-2 - 2\sqrt{3}i + \sqrt{3}i|^2 = |-2 - \sqrt{3}i|^2 = (-2)^2 + (-\sqrt{3})^2 = 4 + 3 = 7\).
Total Sum \(= 31 + 7 = 38\).

Step 4: Final Answer:
The value of the sum is 38.