Circle from an affine complex-modulus equation
= Circle from an affine complex-modulus equation
An equation $|az+b\overline z|=ux+v$, with $z=x+iy$ and real coefficients, becomes $(a+b)^2x^2+(a-b)^2y^2=(ux+v)^2$ after squaring. If $(a+b)^2-u^2=(a-b)^2>0$, completing the square gives a <circle>. The original locus is only the part where $ux+v\geq0$. Checking this sign on the candidate <circle> prevents extraneous points introduced by squaring. This method combines the <complex modulus> with elementary conic geometry.