Arguments after rotation and reflection

A complex number is given by z=a+(a−3)iz=a+(a-3)i, where aa is a positive real constant. It is given that arg⁡z=θ\arg z=\theta, where −π/2<θ<0-\pi/2<\theta<0. Find, leaving answers in terms of θ\theta:

(i) arg⁡(−2z)\arg(-2z) marks;

(ii) arg⁡(z−2a)\arg(z-2a) marks.

[modified 2023/HCI/Prelim/II/3a]

Total marks: 4

Solution

(i)

Argument of a real multiple

z=a+(a−3)iz=a+(a-3)i with a>0a>0 and −π2<arg⁡z=θ<0-\frac\pi2<\arg z=\theta<0. Find arg⁡(−2z)\arg(-2z).

Explore this idea

Hint

  1. Locate zz: it is in quadrant IV, because 0<a<30<a<3.

  2. A negative real multiple reverses the direction.

Answer · (i)

π+θ\pi+\theta.

Explore the idea

A half-turn and a stretch

Move aa. The point −2z-2z is always opposite zz, twice as far from the origin.

Value of a1
Try an example
θ = arg z
-63.4°
arg(−2z)
116.6°
180° + θ
116.6°
  • Moves as the slider changes; a faint path shows where it goes
Key idea

Multiplying by a negative real number reverses direction: add or subtract π\pi to land in the principal range.

Ideas to try
Watch the third readout. It always equals the second, so arg⁡(−2z)=π+θ\arg(-2z)=\pi+\theta.
(ii)

Argument after a real shift

Find arg⁡(z−2a)\arg(z-2a) in terms of θ\theta.

Explore this idea

Hint

  1. Compute z−2az-2a and find its quadrant.

  2. Use the reference angle and the mirror symmetry.

Answer · (ii)

−π−θ-\pi-\theta.

Principal arguments in (−π,π].

Explore the idea

Shifting left by 2a

z−2az-2a sits level with zz but reflected across the imaginary axis. Slide aa and compare the two angles.

Value of a1
Try an example
θ = arg z
-63.4°
arg(z − 2a)
-116.6°
−180° − θ
-116.6°
  • Moves as the slider changes; a faint path shows where it goes
Key idea

Subtracting 2a2a reflects zz across the imaginary axis, here changing the sign of the real part but not the imaginary part.

Ideas to try
Compare the second and third readouts as aa changes. They always match.
Takeaway

Quadrant first, then angle

Decide which quadrant a point is in before stating its argument. Then express the angle relative to the real axis.