## Loci

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 exampleis satisfied by all points of the formWe can often find a Cartesian equation for the set of points satisfying an equation with complex variables by substitutingand manipulating.

Example: Find the set of points satisfying

Substituteto obtain

Collecting real and complex components in each modulus gives

Thenso

Expanding givesand cancellation giveswhich simplifies towhich is the equation of a line.

Example: Find the set of points satisfying

Substituteto obtain

Collecting real and complex components in each modulus gives

Thenso

Expanding givesand 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