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Math Help - Complex Number Question.

  1. #1
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    Complex Number Question.

    The vertices O, A, B of a triangle in the Argand diagram are the points corresponding to the numbers 0, 1 , 1 + i respectively. Show that when the point z describes the perimeter of the triangle OAB, then locus of the points  z^2 consists of part of the real axis part of a parabola and part of the imaginary axis. Sketch this locus.

    [FP3 review exercise question 51]
    I barely understand what this question is asking. excuse my ignorance if this is totally wrong, but isn't z just constant ? as it is the perimeter of a fixed triangle. I am a bit lost on this.


    Bobak
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  2. #2
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    Quote Originally Posted by bobak View Post
    I barely understand what this question is asking. excuse my ignorance if this is totally wrong, but isn't z just constant ? as it is the perimeter of a fixed triangle. I am a bit lost on this.


    Bobak
    It's a poorly worded question alright. Here's my take on it:

    Consider the values of z along the side OA:
    z = a + 0i, 0 < a < 1,
    => z^2 = a^2, 0 < a < 1. These points obviously lie on part of the real axis. The part from 0 to 1.

    Consider the values of z along the side OB:
    z = a + ai, 0 < a < 1,
    => z^2 = 2ai, 0 < a < 1. These points obviously lie on part of the imaginary axis. The part from 0 to 2i.

    Consider the values of z along the side AB:
    z = 1 + ai, 0 < a < 1,
    => z^2 = (1 + ai)^2 = 1 - a^2 + 2ai, 0 < a < 1. So:

    x = 1 - a^2 .... (1)

    y = 2a => y/2 = a .... (2)

    Treat (1) and (2) as parametric equations and eliminate a by substituting (2) into (1):

    x = 1 - (y/2)^2, which is the equation of a sideways parabola. The domain is 0 < x < 1. The range is 0 < y < 2.


    So as z TRACES the perimeter of the triangle, the locus of z^2 does indeed consist of part of the real axis, part of a parabola and part of the imaginary axis. The sketch of this locus should be straightforward.
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  3. #3
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    Thanks a lot Mr F. the locus is a slice of cheese brilliant!
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  4. #4
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    Quote Originally Posted by bobak View Post
    Thanks a lot Mr F. the locus is a slice of cheese brilliant!
    Or a piece of cake lol!
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