A real conjugate sum has two directions

w=−1−i3w=-1-i\sqrt3 and z∗+z≠0z^*+z\ne0. P and Q represent ww and z∗+zz^*+z respectively. Find arg⁡w\arg w exactly. Hence state possible values of ∠POQ\angle POQ, where O is the origin. marks

[2020/VJC/Prelim/II/3a]

Solution

Argument of w and the angle POQ

Principal argument and angle between rays

w=−1−i3w=-1-i\sqrt3 and z∗+z≠0z^*+z\ne0. PP and QQ represent ww and z∗+zz^*+z. Find arg⁡w\arg w exactly and the possible values of ∠POQ\angle POQ.

Explore this idea

Hint

  1. Locate ww in its quadrant and find the reference angle.

  2. Describe QQ using z∗+z=2xz^*+z=2x.

  3. Find the angle for each side.

Answer · Argument of w and the angle POQ

arg⁡w=−2π/3\arg w=-2\pi/3; ∠POQ=2π/3\angle POQ=2\pi/3 if Re⁡z>0\operatorname{Re}z>0, or π/3\pi/3 if Re⁡z<0\operatorname{Re}z<0.

Explore the idea

Q slides along the real axis

Move Q=z∗+z=2xQ=z^*+z=2x along the real axis. The angle at the origin jumps between two values as QQ crosses it.

Real part x of z1.525
Try an example
Q
3.05
angle POQ
120°
side
positive axis: 2π/3
  • Moves as the slider changes; a faint path shows where it goes
Key idea

Because z∗+zz^*+z is real, the angle depends only on the sign of Re⁡z\operatorname{Re}z, giving exactly two possible values.

Ideas to try
Move QQ from the right to the left of the origin and watch the angle change from 120∘120^\circ to 60∘60^\circ.
Takeaway

A sum can be forced onto an axis

Anything of the form z+z∗z+z^* is real, and z−z∗z-z^* is purely imaginary. Use that to place points on an axis.