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Geometric meaning of |z1 – z2|

|z1z2| = distance between the complex numbers z1 and z2 or it is distance of ±(z1z2) from the origin.

Section formula

If C(z) divides the segment AB internally in the ratio of
m: n, then z = 59839.png.
 
If C(z) divides the segment AB externally in the ratio of
m: n, then z = 59833.png.

Condition for collinearity

If there are three real numbers (other than 0) l, m, and n such that lz1 + mz2 + nz3 = 0 and l + m + n = 0 then complex numbers z1, z2, and z3 will represent collinear points.

Standard loci in the argand plane

If z is a variable point and z1, z2 are two fixed points in the argand plane, then
  1. |zz1| = |zz2| Distance of z from two fixed points z1 and z2 is same.
     
    Locus of z is the perpendicular bisector of the line segment joining z1 and z2.
  2. |zz1| + |zz2| = constant (>|z1z2|)
     
    Locus of z is an ellipse (as in ellipse SP + SP = 2a, where S, S′ are foci, P is any point on ellipse and a is semi-major axis).
  3. |zz1| + |zz2| = |z1z2| Locus of z is the line segment joining z1 and z2.
  4. ||zz1| – |zz2|| = |z1z2| Locus of z is a straight line joining z1 and z2 but z does not lie between z1 and z2.
  5. |zz1| – |zz2| = constant (<|z1 z2|) Locus of z is a hyperbola (as in hyperbola SPSP = 2a, where S, S′ are foci, P is any point on hyperbola and a is a semi-transverse axis).
  6. |zz1|2 + |zz2|2 = |z1z2|2. Locus of z is a circle with z1 and z2 as the extremities of diameter.
  7. |zz1| = k|zz2|, (k ≠ 1) Locus of z is a circle.
  8. arg 59825.png = a(fixed) Locus of z is a segment of circle.
  9. arg 59819.png = ±π/2 Locus of z is a circle with z1 and z2 as the vertices of diameter.
  10. arg 59813.png = 0 or π Locus of z is a straight line passing through z1 and z2.

Rotation formula

Let z1, z2, and z3 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 z2z1 and Q is z3z1 and 59807.png (cos α + i sin α) = 59800.png. eiα
 
59794.png




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