Often it is the case that an equation involving complex numbers us satisfied by a whole range of points, often a continuum, called a locus. For example
is satisfied by all points of the form
We can often find a Cartesian equation for the set of points satisfying an equation with complex variables by substituting
and manipulating.
Example: Find the set of points satisfying
Substituteto obtain
Collecting real and complex components in each modulus gives
Then
so
Expanding gives
and cancellation gives
which simplifies to
which is the equation of a line.
Example: Find the set of points satisfying
Substituteto obtain
Collecting real and complex components in each modulus gives
Then
so
Expanding gives
and cancellation gives
We can divide by 3 and complete the square separately for x and y to give
This is the equation of a circle, centre