Complex Numbers
Quadratic equations with unimodular roots
MJAT_TS5_P2
Grade 12
Question:
Let $a,b,c$ be distinct nonzero complex numbers with $|a|=|b|=|c|$. If each of the equations $az^2+bz+c=0$ and $bz^2+cz+a=0$ has a root with modulus 1, then which is/are correct?
A) $b^2=ac$, $c^2=ab$
B) $a^2+b^2+c^2=0$
C) $(a-b)^2+(b-c)^2+(c-a)^2=0$
D) $a-b=b-c=c-a$
Step-by-Step Solution
Key Concept: If $|z|=1$ is a root of $az^2+bz+c=0$: multiply through by $\bar{z}^2$: $a|z|^2/z^2\cdot z^4+\ldots$. For $|z|=1$, $z\bar{z}=1$. Taking conjugate of $az^2+bz+c=0$: $\bar{a}\bar{z}^2+\bar{b}\bar{z}+\bar{c}=0$. With $|a|=|b|=|c|$: $a\bar{a}=b\bar{b}=c\bar{c}=R^2$. Substituting $\bar{z}=1/z$: $\bar{a}/z^2+\bar{b}/z+\bar{c}=0$, multiply by $z^2$: $\bar{a}+\bar{b}z+\bar{c}z^2=0$.
A ✓, C ✓ (from the derived conditions), D ✓. B ✗. Answer: A, C, D.
Correct Answer: ACD