Key Concepts and Formulas

• Equilateral Triangle Condition (Complex Plane): If $z_1, z_2, z_3$ form an equilateral triangle, then $z_1^2 + z_2^2 + z_3^2 = z_1z_2 + z_2z_3 + z_3z_1$. When one vertex is at the origin ($z_3 = 0$), this simplifies to $z_1^2 + z_2^2 = z_1z_2$, which can be further expressed as $(z_1 + z_2)^2 = 3z_1z_2$.
• Vieta's Formulas: For a quadratic equation $Ax^2 + Bx + C = 0$, the sum of the roots is $-\frac{B}{A}$ and the product of the roots is $\frac{C}{A}$.
• Absolute Value: The absolute value of a number $x$, denoted by $|x|$, is its distance from zero. $|x| = x$ if $x \geq 0$, and $|x| = -x$ if $x < 0$.

Step-by-Step Solution

Step 1: Apply the Equilateral Triangle Condition

Since $z_1, z_2,$ and the origin (0) form an equilateral triangle, we can use the equilateral triangle condition with $z_3 = 0$: $z_1^2 + z_2^2 + 0^2 = z_1z_2 + z_2(0) + 0z_1$ This simplifies to: $z_1^2 + z_2^2 = z_1z_2$ To relate this to the sum and product of the roots, we rewrite the left side: $(z_1 + z_2)^2 - 2z_1z_2 = z_1z_2$ Rearranging the terms, we get: $(z_1 + z_2)^2 = 3z_1z_2$ This equation relates the sum and product of the roots for this specific equilateral triangle configuration.

Step 2: Apply Vieta's Formulas

The given quadratic equation is $z^2 + az + 12 = 0$. Applying Vieta's formulas, we have:

• Sum of the roots: $z_1 + z_2 = -\frac{a}{1} = -a$
• Product of the roots: $z_1z_2 = \frac{12}{1} = 12$ These formulas express the sum and product of the roots in terms of the coefficient 'a'.

Step 3: Substitute Vieta's Formulas into the Equilateral Triangle Condition

Substitute $z_1 + z_2 = -a$ and $z_1z_2 = 12$ into the equation $(z_1 + z_2)^2 = 3z_1z_2$: $(-a)^2 = 3(12)$ $a^2 = 36$

Step 4: Solve for 'a'

Taking the square root of both sides: $a = \pm \sqrt{36}$ $a = \pm 6$

Step 5: Find the Value of |a|

The question asks for the value of $|a|$. $|a| = |\pm 6|$ $|a| = 6$

Common Mistakes & Tips

• Sign Errors: Pay close attention to signs when applying Vieta's formulas, especially when identifying the coefficients A, B, and C.
• Forgetting Absolute Value: Remember to take the absolute value at the end. If $a = \pm x$, the absolute value is $|a| = x$.
• Condition for Equilateral Triangle: Memorize the condition $z_1^2 + z_2^2 + z_3^2 = z_1z_2 + z_2z_3 + z_3z_1$ or its equivalent forms, especially when one vertex is the origin.

Summary

We applied the condition for an equilateral triangle in the complex plane, using the origin as one of the vertices. We then used Vieta's formulas to relate the sum and product of the roots to the coefficient 'a' in the given quadratic equation. Substituting these into the equilateral triangle condition allowed us to solve for $a^2$ and subsequently find $|a|$. The final answer is 6.

Final Answer The final answer is \boxed{6}.