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Math Help - Complex and Geometry

  1. #1
    Super Member dhiab's Avatar
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    Complex and Geometry

    What does equation indicate on the plane?
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  2. #2
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    Hello, dhiab!

    What does this equation indicate on the plane: . z + \overline{z} \:=\:k|z|

    . . where: . z \:=\:x + iy,\quad \overline{z} \:=\:x - iy, \quad k \in \mathbb{R}
    The left side is: . z + \overline{z} \:=\:(x+iy) + (x - iy) \:=\:2x

    The right side is: . k|z| \:=\:k\sqrt{x^2+y^2}


    We have: . 2x \:=\:k\sqrt{x^2+y^2}

    Square both sides: . 4x^2 \:=\:k^2x^2 + k^2y^2 \quad\Rightarrow\quad 4x^2 - k^2x^2 \:=\:k^2y^2

    . . (4-k^2)x^2 \:=\:k^2y^2 \quad\Rightarrow\quad \frac{4-k^2}{k^2}x^2 \:=\:y^2

    And we have: . y \;=\;\pm\frac{\sqrt{4-k^2}}{k}\,x


    It is a pair of intersecting lines.

    . . Note that: . |k| \:\leq\: 2\:\text{ and }\:k \:\neq\: 0

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  3. #3
    Super Member dhiab's Avatar
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    Quote Originally Posted by Soroban View Post
    Hello, dhiab!

    The left side is: . x+iy) + (x - iy) \:=\:2x" alt="z + \overline{z} \:=\x+iy) + (x - iy) \:=\:2x" />

    The right side is: . k|z| \:=\:k\sqrt{x^2+y^2}


    We have: . 2x \:=\:k\sqrt{x^2+y^2}

    Square both sides: . 4x^2 \:=\:k^2x^2 + k^2y^2 \quad\Rightarrow\quad 4x^2 - k^2x^2 \:=\:k^2y^2

    . . (4-k^2)x^2 \:=\:k^2y^2 \quad\Rightarrow\quad \frac{4-k^2}{k^2}x^2 \:=\:y^2

    And we have: . y \;=\;\pm\frac{\sqrt{4-k^2}}{k}\,x


    It is a pair of intersecting lines.

    . . Note that: . |k| \:\leq\: 2\:\text{ and }\:k \:\neq\: 0

    HELLO : Thank you
    I'think x, k for same sign , you have the segment is not lines .
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