Question

Let z₁, z₂ be the roots of the equation z² + a z + 12 = 0 and z₁, z₂ form an equilateral triangle with origin. Then, the value of |a| is ________.

Hint:

For any equilateral triangle with vertices $z_1, z_2, z_3$, the relation is $z_1^2 + z_2^2 + z_3^2 = z_1 z_2 + z_2 z_3 + z_3 z_1$.

Answer 1

Correct Answer: 6

Step 1: For an equilateral triangle formed by $z_1, z_2$ and the origin ($z_3 = 0$), the condition is: $z_1^2 + z_2^2 + 0^2 = z_1 z_2 + z_2(0) + (0)z_1$.
Step 2: This simplifies to $z_1^2 + z_2^2 = z_1 z_2$.
Step 3: We can rewrite this as $(z_1 + z_2)^2 - 2z_1 z_2 = z_1 z_2 \implies (z_1 + z_2)^2 = 3z_1 z_2$.
Step 4: From the equation $z^2 + az + 12 = 0$, we have $z_1 + z_2 = -a$ and $z_1 z_2 = 12$.
Step 5: Substituting these values: $(-a)^2 = 3(12) \implies a^2 = 36$.
Step 6: $|a| = \sqrt{36} = 6$.