Question:
If z1, z2 and z3, z4 are two pairs of conjugate complex numbers, then find arg(z1/z4) + arg(z2/z3).
Solution:
According to the question,
We have,
z1 and z2 are conjugate complex numbers.
The negative side of the real axis
= r1 (cos θ1 – i sin θ1)
= r1 [cos (-θ1) + I sin (-θ1)]
Similarly, z3 = r2 (cos θ2 – i sin θ2)
⇒ z4 = r2 [cos (-θ2) + I sin (-θ2)]
$\Rightarrow \arg \left(\frac{\mathrm{z}_{1}}{\mathrm{z}_{4}}\right)+\arg \left(\frac{\mathrm{z}_{2}}{\mathrm{z}_{3}}\right)=\arg \left(\mathrm{z}_{1}\right)-\arg \left(\mathrm{z}_{4}\right)+\arg \left(\mathrm{z}_{2}\right)-\arg \left(\mathrm{z}_{3}\right)$
= θ1 – (-θ2) + (-θ1) – θ2
= θ1 + θ2 – θ1 – θ2
= 0
⇒ arg(z1/z4) + arg(z2/z3) = 0
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All Study Material
- JEE Main
- Exam Pattern
- Previous Year Papers
- PYQ Chapterwise
- Physics
- Kinematics 1D
- Kinemetics 2D
- Friction
- Work, Power, Energy
- Centre of Mass and Collision
- Rotational Dynamics
- Gravitation
- Calorimetry
- Elasticity
- Thermal Expansion
- Heat Transfer
- Kinetic Theory of Gases
- Thermodynamics
- Simple Harmonic Motion
- Wave on String
- Sound waves
- Fluid Mechanics
- Electrostatics
- Current Electricity
- Capacitor
- Magnetism and Matter
- Electromagnetic Induction
- Atomic Structure
- Dual Nature of Matter
- Nuclear Physics
- Radioactivity
- Semiconductors
- Communication System
- Error in Measurement & instruments
- Alternating Current
- Electromagnetic Waves
- Wave Optics
- X-Rays
- All Subjects
- Physics
- Motion in a Plane
- Law of Motion
- Work, Energy and Power
- Systems of Particles and Rotational Motion
- Gravitation
- Mechanical Properties of Solids
- Mechanical Properties of Fluids
- Thermal Properties of matter
- Thermodynamics
- Kinetic Theory
- Oscillations
- Waves
- Electric Charge and Fields
- Electrostatic Potential and Capacitance
- Current Electricity
- Thermoelectric Effects of Electric Current
- Heating Effects of Electric Current
- Moving Charges and Magnetism
- Magnetism and Matter
- Electromagnetic Induction
- Alternating Current
- Electromagnetic Wave
- Ray Optics and Optical Instruments
- Wave Optics
- Dual Nature of Radiation and Matter
- Atoms
- Nuclei
- Semiconductor Electronics: Materials, Devices and Simple Circuits.
- Chemical Effects of Electric Current,
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