Root midpoint and a parallelogram

The complex number zz satisfies z2+(2−4i)z=cz^2+(2-4i)z=c, where cc is a complex constant.

(i) Given z1,z2z_1,z_2 are two distinct roots, show (z1+z2)/2=−1+2i(z_1+z_2)/2=-1+2i. marks

It is given c=4ic=4i and Im⁡(z1)<Im⁡(z2)\operatorname{Im}(z_1)<\operatorname{Im}(z_2).

(ii) Find z1,z2z_1,z_2 in the form x+iyx+iy, with x,yx,y real. Leave x,yx,y exact. marks

(iii) On an Argand diagram indicate clearly P1,P2P_1,P_2 representing z1,z2z_1,z_2 respectively. marks

(iv) O and Q represent 0 and ww respectively. OP1P2QOP_1P_2Q is a parallelogram. Using (iii), find ww. marks

[2022/VJC/Promo/9]

Total marks: 9

Solution

(i)

Sum of roots

z2+(2−4i)z=cz^2+(2-4i)z=c has distinct roots z1,z2z_1,z_2. Show that z1+z22=−1+2i\frac{z_1+z_2}2=-1+2i.

Explore this idea

Hint

  1. Write the equation as z2+(2−4i)z−c=0z^2+(2-4i)z-c=0 and use the sum of roots.

Answer · (i)

(z1+z2)/2=−1+2i(z_1+z_2)/2=-1+2i.

Explore the idea

The roots rotate around a fixed midpoint

Let c=kic=ki and slide kk. The roots move, but their midpoint stays at −1+2i-1+2i.

Value of k in c = k i4
Try an example
z₁
-1 + 0.268i
z₂
-1 + 3.732i
midpoint
-1 + 2i
  • Moves as the slider changes; a faint path shows where it goes
Key idea

The constant cc changes how far apart the roots are and the direction between them, but never their midpoint.

Ideas to try
  1. Slide kk and watch the readout of the midpoint. It never changes.
  2. At k=4k=4 the roots are on one vertical line, the case in part (ii).
(ii)

Solve with a complex discriminant

With c=4ic=4i and Im⁡z1<Im⁡z2\operatorname{Im}z_1<\operatorname{Im}z_2, find z1,z2z_1,z_2 in the form x+iyx+iy with exact x,yx,y.

Hint

  1. Compute the discriminant of z2+(2−4i)z−4i=0z^2+(2-4i)z-4i=0.

  2. Use z=−1+2i±D2z=-1+2i\pm\frac{\sqrt D}2.

Answer · (ii)

z1=−1+(2−3)iz_1=-1+(2-\sqrt3)i, z2=−1+(2+3)iz_2=-1+(2+\sqrt3)i.

(iii)

Plot the roots

On an Argand diagram indicate P1,P2P_1,P_2 representing z1,z2z_1,z_2.

Explore this idea

Hint

  1. Read off the coordinates and plot them.

The ordered parallelogram OP₁P₂Q
The ordered parallelogram OP₁P₂QScroll across to see the full diagram.

Answer · (iii)

P1=(−1,2−3)P_1=(-1,2-\sqrt3), P2=(−1,2+3)P_2=(-1,2+\sqrt3).

Explore the idea

Two points on a vertical line

Drag to look around the Argand plane. Real parts run along the horizontal axis and imaginary parts along the vertical axis.

Reset the controls
Key idea

A negative real discriminant means the roots differ by an imaginary number: they stack vertically.

Ideas to try
The separation of the roots is 23≈3.462\sqrt3\approx3.46, and the midpoint is the red point.
(iv)

Parallelogram on the Argand diagram

OO and QQ represent 00 and ww, and OP1P2QOP_1P_2Q is a parallelogram. Find ww.

Explore this idea

Hint

  1. Opposite sides of a parallelogram are equal vectors: OQ→=P1P2→\overrightarrow{OQ}=\overrightarrow{P_1P_2}.

Answer · (iv)

w=23iw=2\sqrt3i.

Explore the idea

A parallelogram with O at one corner

Drag to look around the Argand plane. Real parts run along the horizontal axis and imaginary parts along the vertical axis.

Reset the controls
Key idea

A parallelogram OP1P2QOP_1P_2Q with a vertex at 00 gives w=z2−z1w=z_2-z_1: the side vectors just add and subtract.

Ideas to try
OP1OP_1 and QP2QP_2 are parallel and equal; OQOQ and P1P2P_1P_2 are parallel and equal.
Takeaway

Sums fix the centre

The sum of the roots depends only on the coefficient of zz. Roots then spread out symmetrically about half that sum.