Locus of a complex number

Question

Let $f$ be a mapping of ℂ in ℂ and M be a point of affix $z= x+iy$. Find the set of points M, such that:$f(z)$ is a pure imaginary number and $f(z)=z^2 + z + 1$. I'm new to complex numbers but what I can imply from this question is that $z^2 + z + 1$ is a point that lies on the y axis ( in this case the imaginary axis). So I put $z^2 + z +1 = y$. But again, I'm new to complex numbers and I think I'm messing up with the answer. Please help. Thanks.

Answer 1

Since $f(z)$ is purely imaginary, $f(z) = 0+iq$, where $q$ is some real number.

Replacing $z=x+iy$ in $f(z)$ and equating the real part to zero gives the locus of the points $(x,y)$.

$f(x+iy)=(x^2-y^2+x+1)+i(2xy+y)=0+iq$.

$\implies x^2-y^2+x+1=0$ or $y^2=x^2+x+1$.