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Thread: Complex and Geometry

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    Super Member dhiab's Avatar
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    Complex and Geometry

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

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

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

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


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

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

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

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


    It is a pair of intersecting lines.

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

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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: .$\displaystyle z + \overline{z} \:=\x+iy) + (x - iy) \:=\:2x$

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


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

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

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

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


    It is a pair of intersecting lines.

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

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