Complex Numbers

Applications and Advanced Problems

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Applications and Advanced Problems

Applications and Advanced Problems

This lesson covers the following key concepts:

  • Triangle inequality: |z₁ + z₂| ≤ |z₁| + |z₂|
  • Maximum and minimum values of |z|
  • Complex number equations and locus
  • Transformation of complex numbers
  • Rotation and scaling in complex plane
  • Applications to geometry problems
  • JEE-specific problem patterns

Important Formulas

  • ∣z1+z2∣≤∣z1∣+∣z2∣ (triangle inequality)|z_1 + z_2| \leq |z_1| + |z_2| \text{ (triangle inequality)}
  • ∣∣z1∣−∣z2∣∣≤∣z1−z2∣||z_1| - |z_2|| \leq |z_1 - z_2|
  • ∣z1+z2∣2+∣z1−z2∣2=2(∣z1∣2+∣z2∣2)|z_1 + z_2|^2 + |z_1 - z_2|^2 = 2(|z_1|^2 + |z_2|^2)
  • Rotation by α:z′=zeiα\text{Rotation by } \alpha: z' = ze^{i\alpha}
  • Scaling: z′=kz where k∈R\text{Scaling: } z' = kz \text{ where } k \in \mathbb{R}
  • ∣z−z1∣+∣z−z2∣=k (ellipse if k>∣z1−z2∣)|z - z_1| + |z - z_2| = k \text{ (ellipse if } k > |z_1 - z_2|)
  • ∣∣z−z1∣−∣z−z2∣∣=k (hyperbola)||z - z_1| - |z - z_2|| = k \text{ (hyperbola)}