Loci of Complex Numbers in the Argand Plane

A very useful skill is translating a condition written in terms of $z$ (such as $|z-1|=3$) into the familiar Cartesian equation of a curve — and conversely, recognising standard curves when they appear disguised as complex equations. The general method is always the same: substitute $z = x+iy$, simplify using $|a+ib|=\sqrt{a^2+b^2}$ and the conjugate rules, and read off the resulting equation in $x,y$.

A few standard loci come up repeatedly and are worth knowing by pattern:

• Straight line. The general equation of a circle in the complex plane can be written $z\bar z + \bar b z + b\bar z + c = 0$ with $b\in\mathbf C$, $c\in\mathbf R$ — substituting $z=x+iy$ recovers the familiar $x^2+y^2+2gx+2fy+c=0$. A degenerate/limiting case of this family, or a condition like $\operatorname{Re}(z) = k$ or $\operatorname{Arg}(z-z_0)=$ constant, gives a straight line instead of a circle.
• Circle as a diameter condition. $\operatorname{Arg}\!\left(\dfrac{z-z_1}{z-z_2}\right) = \pm\dfrac{\pi}{2}$ describes the circle having the segment joining $z_1$ and $z_2$ as a diameter — because an angle inscribed in a semicircle is always a right angle (Thales' theorem), so any point $z$ on that circle sees $z_1$ and $z_2$ at a right angle. Equivalently, $|z-z_1|^2 + |z-z_2|^2 = |z_1-z_2|^2$.
• Circle or line from a ratio. For $\left|\dfrac{z-z_1}{z-z_2}\right| = k$ (with $z_1\ne z_2$ fixed): the locus is a circle if $k\ne 1$, and a straight line (specifically, the perpendicular bisector of $z_1z_2$) if $k=1$ — since $k=1$ means $z$ is equidistant from $z_1$ and $z_2$.
• Ellipse. $|z-z_1| + |z-z_2| = 2a$ (with $2a > |z_1-z_2|$) is an ellipse with foci at the points $z_1, z_2$ — the direct complex-number translation of the 'sum of distances to two foci is constant' definition of an ellipse.
• Hyperbola. $\big||z-z_1| - |z-z_2|\big| = 2a$ (with $0 < 2a < |z_1-z_2|$) is a hyperbola with foci $z_1, z_2$, by the same logic applied to the difference of distances. …