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Put $z=x+iy$. Squaring the two nonnegative moduli and expanding gives $x^2+(y+3{)}^2=9(x^2+y^2)$. Completing the square yieldsThus the locus is an Apollonius circle with center $3i/8$ and radius $9/8$. No extraneous points were introduced by squaring, because both original sides are nonnegative. Its intersections with the imaginary axis are $-3i/4$ and $3i/2$, useful landmarks for the sketch. It surrounds zero but does not pass through zero:

Take the wire at $(a,0)$ to carry $+\lambda$ and that at $(-a,0)$ to carry $-\lambda$. Ifthen, up to one additive constant,Thus an equipotential with $k=2\pi {\epsilon }_0\varphi /\lambda$ satisfies $r_{-}/r_{+}=e^k$. For $k\ne 0$, completing the square gives the Apollonius circleIts centre is $(a\coth k,0)$ and its radius is $a/|\sinh k|$. Positive and negative values give nested circles around the positive and negative wires. The electric field is orthogonal to these circles and points from the positive wire toward the negative wire. For $\varphi =0$, the limiting equipotential is the straight line $x=0$.

Direct superposition givesIn the limit $a\to 0$ with $\lambda a=p$, the potential and field become those of a two-dimensional electric dipole:and