Zeros of f(z) = (az-b)npm (cz-d)n

The zeros of
f(z) = (az - b)^{n} \pm (cz - d)^{n}
are found to lie on a circle of radius
|(ad - cb)/(|a|^{2} - |c|^{2})|
with its center at
z = (a^{\ast}b - c^{\ast}d)/(|a|^{2} - |c|^{2})
, where
a, b, c
, and
d
are complex numbers and
n
is assumed real. When
|a| = |c|
the locus of the zeros is a straight line perpendicular to the line joining the points
b/a
and
b/c
and intersecting it at
z = 0.5(b/a + d/c)
. The zeros are found analytically and constructed geometrically.