Five points from one polar vector

zz has modulus rr and argument θ\theta, with 0<r<10<r<1 and π/2<θ<π\pi/2<\theta<\pi. ww satisfies ∣w∣=∣z∣|w|=|z| and arg⁡w+arg⁡z=π\arg w+\arg z=\pi.

A, B, C, D and E represent z,w,z−w,z∗,−izz,w,z-w,z^*,-iz respectively, where z∗z^* is the conjugate of zz. On a single Argand diagram illustrate all five points. marks

[modified 2019/NJC/Prelim/I/1]

Solution

Plot A, B, C, D, E

Reflect, rotate and subtract

zz has modulus rr and argument θ\theta, with 0<r<10<r<1 and π2<θ<π\frac\pi2<\theta<\pi. Also ∣w∣=∣z∣|w|=|z| and arg⁡w+arg⁡z=π\arg w+\arg z=\pi. Illustrate A,B,C,D,EA,B,C,D,E representing zz, ww, z−wz-w, z∗z^*, −iz-iz.

Explore this idea

Hint

  1. Write z=x+iyz=x+iy with x<0<yx<0<y.

  2. Find each point from xx and yy.

  3. Mark the angles θ\theta and π−θ\pi-\theta.

Five polar transformations with 0<r<1 and π/2<θ<π
Five polar transformations with 0<r<1 and π/2<θ<πScroll across to see the full diagram.

Answer · Plot A, B, C, D, E

With x=rcos⁡θ<0x=r\cos\theta<0, y=rsin⁡θ>0y=r\sin\theta>0: A=(x,y),B=(−x,y),C=(2x,0),D=(x,−y),E=(y,−x)A=(x,y),B=(-x,y),C=(2x,0),D=(x,-y),E=(y,-x).

0<r<10<r<1, π/2<θ<π\pi/2<\theta<\pi; plot represents this family.

Explore the idea

The family of five points

Turn zz through the second quadrant. BB mirrors AA in the imaginary axis, DD mirrors it in the real axis, EE is a quarter-turn clockwise, and CC stays on the real axis.

Argument θ of z130°
Try an example
z
-0.514 + 0.613i
w
0.514 + 0.613i
z − w
-1.028
arg w
50°
  • Moves as the slider changes; a faint path shows where it goes
Key idea

BB is AA reflected in the imaginary axis, so z−wz-w is purely real. EE is AA turned −90∘-90^\circ about the origin.

Ideas to try
  1. At θ=135∘\theta=135^\circ the points AA, BB, DD, EE form a square on the circle, and C=2xC=2x is its far-left diagonal tip.
  2. Slide θ\theta toward 180∘180^\circ: AA approaches (−r,0)(-r,0) and BB approaches (r,0)(r,0), at opposite ends of the real axis. CC approaches (−2r,0)(-2r,0), so z−wz-w approaches 2z2z.
Takeaway

Describe a family, not a single picture

A parameter range gives a family of diagrams. State the signs and relationships that all of them share.