Complex Numbers

Locus and Geometry in Argand Plane

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Locus and Geometry in Argand Plane

Locus and Geometry in Argand Plane

This lesson covers the following key concepts:

  • Distance between two complex numbers
  • Equation of straight line in complex form
  • Equation of circle in complex form
  • Perpendicular bisector of segment joining z₁ and z₂
  • Rotation of complex numbers
  • Collinearity and concurrency conditions
  • Maximum and minimum values of |z| under constraints

Important Formulas

  • ∣z−z0∣=r (circle with center z0 and radius r)|z - z_0| = r \text{ (circle with center } z_0 \text{ and radius } r)
  • ∣z−z1∣=∣z−z2∣ (perpendicular bisector)|z - z_1| = |z - z_2| \text{ (perpendicular bisector)}
  • arg⁡(z−z1z−z2)=θ (locus is arc of circle)\arg\left(\frac{z - z_1}{z - z_2}\right) = \theta \text{ (locus is arc of circle)}
  • ∣z−z1∣+∣z−z2∣=2a (ellipse with foci z1,z2)|z - z_1| + |z - z_2| = 2a \text{ (ellipse with foci } z_1, z_2)
  • Rotation by θ:z′=zeiθ\text{Rotation by } \theta: z' = z e^{i\theta}
  • Three points collinear: arg⁡(z3−z1z2−z1)=0 or π\text{Three points collinear: } \arg\left(\frac{z_3 - z_1}{z_2 - z_1}\right) = 0 \text{ or } \pi