Let ‘z’ be a complex number and ‘a’ be a real parameter such that z^2 + az + a^2 = 0, then

Q: Let ‘z’ be a complex number and ‘a’ be a real parameter such that z2 + az + a2 = 0, then

(A) locus of z is a pair of straight lines

(B) locus of z is a circle

(C) arg(z) = ± 2π/3

(D) |z| =|a| .

Sol. z2 + az + a2 = 0

⇒ z = aw, aw2 ( where ‘w’ is non real root of cube unity )

⇒ locus of z is a pair of straight lines

and arg (z) =arg(a) + arg(w) or arg(a) + arg(w2)

⇒ arg(z) = ± 2π/3

also, |z| = |a||w| or |a| |w2| ⇒ |z| = |a|

Hence (A), (C) and (D) are the correct answers.