complex help (1 Viewer)

[littleboy]

im kinda stuck on a few questions...
so any help would be appreciated

1.The origin O and the points A,B and C represents the complex numbers z , 1/z and z + 1/z respectively are joined to form a quadrilateral. Write down the condition or conditions for z so that the quadrilateral OABC will be a) a rhombus, b) a square

2. Write down the six complex sixth roots of unity in modulus-argument form. Sketch the roots on an Argand diagram and explain why they form a regular hexagon.

 

[pLuvia]

2 Let z⁶ = 1
= cis 0

.: z = cis(2kpi/6) where k = 0, ±1, ±2, 3

When
k = 0, z = cis 0 = 1
k = 1, z = cis pi/3
k = -1, z = cis -pi/3
k = 2, z = cis 2pi/3
k = -2, z = cis -2pi/3
k = 3, z = cis pi = -1

The locus of all points lie on a circle, centre (0,0) and radius 1
Each angle is spread out by pi/6, hence using the cosine rule the sides will be the same

Hence the shape of a rectangle

 

[Trebla]

rectangle

Don't you mean a regular hexagon?

 

[VivianHsu]

1.
say z = r cis@
then 1/z = 1/r cis(-@)
If it's either a square or a rhombus (i.e. all four sides are equal), r=1/r so r=1

In addition,
Rhombus: argument of z can be anything except pi/2 or -pi/2
Square: z has argument pi/4, 3pi/4, -pi/4 or -3pi/4

Also I think the quadrilateral is OACB not OABC.