H2 Math Complex Numbers

• Audi

  27 Feb 12, 19:08

  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

  27 Feb 12, 20:19

  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

  28 Feb 12, 19:05

  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

  29 Feb 12, 10:16

  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.