Question

Let $z_1, z_2 \in \mathbb{C}$ be the distinct solutions of the equation $z^2 + 4z - (1 + 12i) = 0$. Then $|z_1|^2 + |z_2|^2$ is equal to :

• A. 18
• B. 22
• C. 29
• D. 34

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
We need to solve a quadratic equation with complex coefficients. This can be done by completing the square or using the quadratic formula.
: Key Formula or Approach:
The equation is $z^2 + 4z - (1 + 12i) = 0$.
Completing the square: $(z + 2)^2 - 4 - 1 - 12i = 0 \implies (z + 2)^2 = 5 + 12i$.
Step 2: Detailed Explanation:
Let $z + 2 = w$. We need to find the square root of $5 + 12i$.
Let $w = a + ib$. Then $w^2 = a^2 - b^2 + 2abi = 5 + 12i$.
$a^2 - b^2 = 5$ and $2ab = 12 \implies ab = 6$.
Using the identity $(a^2 + b^2)^2 = (a^2 - b^2)^2 + (2ab)^2$:
$(a^2 + b^2)^2 = 5^2 + 12^2 = 25 + 144 = 169 \implies a^2 + b^2 = 13$.
Solving $a^2 - b^2 = 5$ and $a^2 + b^2 = 13$:
$2a^2 = 18 \implies a^2 = 9 \implies a = \pm 3$.
$2b^2 = 8 \implies b^2 = 4 \implies b = \pm 2$.
Since $ab = 6$ (positive), $a$ and $b$ have the same sign.
$w = \pm(3 + 2i)$.
Case 1: $z_1 + 2 = 3 + 2i \implies z_1 = 1 + 2i$.
Case 2: $z_2 + 2 = -3 - 2i \implies z_2 = -5 - 2i$.
Now calculate the moduli:
$|z_1|^2 = 1^2 + 2^2 = 5$.
$|z_2|^2 = (-5)^2 + (-2)^2 = 25 + 4 = 29$.
Sum $= 5 + 29 = 34$.
Step 3: Final Answer:
The sum $|z_1|^2 + |z_2|^2$ is 34.