Question

If \( z = x + iy \) satisfies the equation \[ z^2 + az + a^2 = 0, \quad a \in \mathbb{R}, \] then:

• A. \( |z| = |a| \)
• B. \( |z - a| = a \)
• C. \( z = |a| \)
• D. \( z = a \)
  \

Hint:

For quadratic equations with complex roots, use the quadratic formula and compute the modulus to determine geometric properties of the solutions.

Answer 1

The Correct Option is A

We are given the quadratic equation in \( z \): \[ z^2 + az + a^2 = 0, \quad a \in \mathbb{R}. \] Step 1: Solve for \( z \)
Using the quadratic formula: \[ z = \frac{-a \pm \sqrt{a^2 - 4a^2}}{2}. \] \[ z = \frac{-a \pm \sqrt{a^2 (1 - 4)}}{2}. \] \[ z = \frac{-a \pm \sqrt{-3a^2}}{2}. \] \[ z = \frac{-a \pm i\sqrt{3} |a|}{2}. \] Step 2: Compute \( |z| \)
\[ |z| = \sqrt{\left( \frac{-a}{2} \right)^2 + \left( \frac{\sqrt{3} |a|}{2} \right)^2 }. \] \[ = \sqrt{\frac{a^2}{4} + \frac{3a^2}{4}}. \] \[ = \sqrt{\frac{4a^2}{4}} = \sqrt{a^2}. \] \[ = |a|. \] Step 3: Conclusion
Thus, we obtain: \[ |z| = |a|. \] The correct answer is: \[ \boxed{|z| = |a|}. \] \bigskip