Complex plane locus question (another one)

[applestrudle]

Homework Statement

again there is no answer provided in the book!

a +bt +ct^2 = z where t is a real parameter, and a, b, c are complex numbers with b/c real

Homework Equations

The Attempt at a Solution

b/c real indicates that b and c are pure imaginary so when you split the equation into real and imaginary parts:

x = Re(a)

y = Im(a) + Im(b)t + Im(c)t^2

since t is a parameter, in the complex plane, the locus of z is a straight vertical line with x = Re(a)?

 

[Mugged]

b/c real indicates that b and c are pure imaginary

Are you sure? What if b = 1 + i and c = 2 + 2i ?