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

Concept: First find the complex roots and then evaluate the modulus.
Step 1: Solve the quadratic \[ z^2+4z+16=0 \] \[ (z+2)^2=-12 \] \[ z=-2\pm2\sqrt{3}i \] Thus \[ S=\{-2+2\sqrt3 i,\;-2-2\sqrt3 i\} \]
Step 2: Compute the modulus For \(z_1=-2+2\sqrt3 i\): \[ z_1+\sqrt3 i=-2+3\sqrt3 i \] \[ |z_1+\sqrt3 i|^2=(-2)^2+(3\sqrt3)^2 \] \[ =4+27=31 \] For \(z_2=-2-2\sqrt3 i\): \[ z_2+\sqrt3 i=-2-\sqrt3 i \] \[ |z_2+\sqrt3 i|^2=(-2)^2+(\sqrt3)^2 \] \[ =4+3=7 \]
Step 3: Sum \[ 31+7=38 \] Thus \[ \boxed{38} \]