Complex Numbers can be represented on a Cartesian Plane(called the complex plane or an argand diagram) as a point or Vector where . Therefore the -axis is the real axis and the -axis is the imaginary axis.
Loci
Circle
Given , is the distance between points . So you can replace with the general point to obtain a locus of points that are all equidistant form . Therefore you get a circle centre described by to locus
Circle
The locus of points that satisfy is a circle.
You can find the equation by splitting into then squaring both sides (which gets rid of modulus and terms) and solving from there.
In basic terms it’s the locus of points that are times further from than they are from .
When it’s a straight line perpendicular bisector which can be thought of as a circle with infinite radius
Circle Arc
The Locus is an arc of a circle with end points . The arc is drawn anticlockwise from to . You can find the equation of a circle by considering the 2 congruent isosceles triangles formed by , where is the circle centre and is the midpoint of the line .
Perpendicular Bisector
Given the locus Is the perpendicular bisector of the line segment connecting
Half Line
Given the locus of points described by forms a straight half-line that makes angle with the line extending from that is parallel to the real axis but does not include
Transformations on the Complex Plane
You can transform loci by mapping the plane onto a plane to get represents translation by Vector, , represents enlargement with scale factor centre represents anti-clockwise rotation of about