H2 Math Complex Numbers

Audi SG

In an Argand diagram, the points P1 and P2 represents the complex number z and z^2 respectively. It is given that the triangle OP1P2 forms an equilateral triangel where O is the origin. Find the modulus and argument of z. Hence determine the area of triangle OP1P2.

Any one has a clue how I should go about solving this question?

Thank you!

wee_ws SG

Hi,

Draw a diagram to represent the points P_1 and P_2.

P_1 has argument alpha and length k, P_2 has argument ? and length ? using the facts:

1. |z^n| = |z|^n,

2. arg (z^n) = n arg (z).

Since O P_1 P_2 is an equilateral triangle:

1. OP_1 = OP_2, from which we could conclude the length k,

2. angle P_1 O P_2 = pi/3, from which we could conclude the argument alpha.

Area should be easy to find then.

Thanks.

Cheers,
Wen Shih

wee_ws SG

Hi,

Here is a similar question for you to ponder over:

RI(JC) 2010 Prelim/P1/Q9(b)(ii) with modification to some phrases

In an Argand diagram, the point A represents the complex number 2 - sqrt(3) + i. If A, B and C are the vertices of an equilateral triangle taken in clockwise order, and these three points lie on a circle whose centre is at the origin, find the complex number represented by B in cartesian form p + iq where p, q are exact real numbers to be found.

Thanks.

Cheers,
Wen Shih

eagle SG

I was thinking of this solution without any use of diagrams

let z = re^it, then z^2 = r^2 e^i2t

so since it is an equiliateral triangle, |z| = |z^2|

or r = r^2
so r (r-1) = 0
r = 0 (reject) or r = 1 (ans)

Also for argument,

arg(z) = t and arg(z^2) = 2t,
The difference in arguments should be pi/3 and the arg(z) can then be found.