Multiple choice

lf $z_{1},\ z_{2}$ are roots of equation $z^{2}-az+a^{2}=0$, then $|\displaystyle \frac{z_{1}}{z_{2}}|=$

  1. $a^{2}$

  2. a

  3. 2a

  4. 1
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

For z^2 - az + a^2 = 0, the roots are z = (a +/- sqrt(a^2 - 4a^2))/2 = a(1 +/- i*sqrt(3))/2. The magnitude of both roots is |a| * sqrt(1/4 + 3/4) = |a|. Thus |z1/z2| = |z1|/|z2| = |a|/|a| = 1.

AI explanation

The discriminant of the given equation z^2 - az + a^2 = 0 is D = a^2 - 4a^2 = -3a^2, which means the roots are complex. Because complex roots for this equation come in the form of conjugate pairs when considering the properties of cube roots of unity scaled by magnitude, both roots must have the exact same magnitude. Since the roots have identical magnitudes, the absolute value of their ratio z1 / z2 is 1.

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