Geometric meaning of |z_{1} â€“ z_{2}|
|z_{1} â€“ z_{2}| = distance between the complex numbers z_{1} and z_{2} or it is distance of Â±(z_{1} â€“ z_{2}) from the origin.
Section formula
If C(z) divides the segment AB internally in the ratio of
m: n, then z = .
If C(z) divides the segment AB externally in the ratio of
m: n, then z = .
Condition for collinearity
If there are three real numbers (other than 0) l, m, and n such that lz_{1} + mz_{2} + nz_{3} = 0 and l + m + n = 0 then complex numbers z_{1}, z_{2}, and z_{3} will represent collinear points.
Standard loci in the argand plane
If z is a variable point and z_{1}, z_{2} are two fixed points in the argand plane, then
- |z â€“ z_{1}| = |z â€“ z_{2}| â‡’ Distance of z from two fixed points z_{1} and z_{2} is same.
- |z â€“ z_{1}| + |z â€“ z_{2}| = constant (>|z_{1} â€“ z_{2}|)
- |z â€“ z_{1}| + |z â€“ z_{2}| = |z_{1} â€“ z_{2}| â‡’ Locus of z is the line segment joining z_{1} and z_{2}.
- ||z â€“ z_{1}| â€“ |z â€“ z_{2}|| = |z_{1} â€“ z_{2}| â‡’ Locus of z is a straight line joining z_{1} and z_{2} but z does not lie between z_{1} and z_{2}.
- |z â€“ z_{1}| â€“ |z â€“ z_{2}| = constant (<|z_{1 }â€“ z_{2}|) â‡’ Locus of z is a hyperbola (as in hyperbola Sâ€²P â€“ SP = 2a, where S, Sâ€² are foci, P is any point on hyperbola and a is a semi-transverse axis).
- |z â€“ z_{1}|^{2} + |z â€“ z_{2}|^{2} = |z_{1} â€“ z_{2}|^{2}. â‡’ Locus of z is a circle with z_{1} and z_{2} as the extremities of diameter.
- |z â€“ z_{1}| = k|z â€“ z_{2}|, (k â‰ 1) â‡’ Locus of z is a circle.
- arg = a(fixed) â‡’ Locus of z is a segment of circle.
- arg = Â±Ï€/2 â‡’ Locus of z is a circle with z_{1} and z_{2} as the vertices of diameter.
- arg = 0 or Ï€ â‡’ Locus of z is a straight line passing through z_{1} and z_{2}.
Rotation formula
Let z_{1}, z_{2}, and z_{3} be the three vertices of a triangle ABC described in the counterclockwise sense. Draw OP and OQ parallel and equal to AB and AC, respectively. Then the point P is z_{2} â€“ z_{1} and Q is z_{3} â€“ z_{1} and (cos Î± + i sin Î±) = . e^{iÎ±}