Show that the equation $\displaystyle Az\bar{z}+\bar{B}z+B\bar{z}+C=0$, where $\displaystyle A,C \in \mathbb{R}$ and $\displaystyle B \in \mathbb{C}$, represents a circline. Prove that every circline is representable in this form.
Show that the equation $\displaystyle Az\bar{z}+\bar{B}z+B\bar{z}+C=0$, where $\displaystyle A,C \in \mathbb{R}$ and $\displaystyle B \in \mathbb{C}$, represents a circline. Prove that every circline is representable in this form.
For the first part, let A = 0 and write out the expression with B = b1 + i b2 and z = x + iy. You should be able to write the equation in the standard form of a line. For the second, use this representations and complete the squares to write in the standard form of a circle. For the second, assume a general line (or circle) then work backwards similar to the first part to hit the form required.
That method is a bit pedantic, but works. There is a more elegant approach. For the first, you can use an inner product, but I don't recall the details right now. For the second, you work on the Riemann Sphere which, I'm guessing, you haven't studied that yet ... but you will.