Question

Let z be a complex number and a be a real parameter, such that $z^2+az+a^2=0$. Then

• A. Locus of z is a pair of straight lines
• B. $Arg\left(z\right)=\pm \frac{2\pi }{3}$
• C. Locus of z is a circle
• D. $|z|=|a|$

Answer 1

Answer: (A), (B), (D)

$|z|=|a|$

$z^2+az+a^2=0$

$\Rightarrow z=a\omega ,a{\omega }^2$

(where $\omega$ is non-real complex cube root of unity)

Locus of z is a pair of straight lines

and $Arg\left(z\right)=Arga+Arg\omega \text{or Arg}a+\text{Arg}\left({\omega }^2\right)$

Arg $z=\pm \frac{2\pi }{3}$

Also $|z|=|a||\omega |\text{or}|a||{\omega }^2|\Rightarrow |z|=|a|$