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Thread: Complex Locus

  1. #1
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    Complex Locus

    z = x + iy, which satisfies

    z^2 + z^ -2 = 8

    Find the locus of P in terms of x and y.

    Is the general aproach to get a equantion in x and y?

    If so, i tried this and ive got z^4 - 8z^2 + 1 = 0
    Do i just sub in x + iy as z and expand to get locous?

    If not what do i do?
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  2. #2
    Super Member Failure's Avatar
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    Quote Originally Posted by Lukybear View Post
    z = x + iy, which satisfies

    z^2 + z^ -2 = 8

    Find the locus of P in terms of x and y.

    Is the general aproach to get a equantion in x and y?

    If so, i tried this and ive got z^4 - 8z^2 + 1 = 0
    Do i just sub in x + iy as z and expand to get locous?
    You could do that, but it's likely to be rather messy.
    If not what do i do?
    I would suggest substituting $\displaystyle w := z^2$ first. If you do that you'll find that $\displaystyle w_{1,2}=4\pm \sqrt{15}$ are the only solutions. Now solve the two equations $\displaystyle (x+iy)^2=4\pm \sqrt{15}$ and you are done.
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  3. #3
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    Hello, Lukybear!

    $\displaystyle z \,=\, x + iy\text{, which satisfies: }\;z^2 + z^{-2} \:=\: 8$

    $\displaystyle \text{Find the locus of }P\text{ in terms of }x\text{ and }y.$

    The problem is not clearly stated.
    I assume that we want the locus of $\displaystyle z.$

    We have: .$\displaystyle z^4 - 8z^2 + 1 \:=\:0$

    Quadratic Formula: .$\displaystyle z^2 \:=\:\dfrac{8\pm\sqrt{60}}{2} \:=\:\dfrac{8\pm2\sqrt{15}}{2} \:=\:4 \pm\sqrt{15}$ .[1]


    We note that: .$\displaystyle (\sqrt{5}\pm \sqrt{3})^2 \;=\;5 \pm 2\sqrt{15} + 3 \;=\;8 \pm 2\sqrt{15} \;=\;2(4 \pm \sqrt{15})$

    . . Hence: .$\displaystyle 4 \pm \sqrt{15} \:=\:\dfrac{(\sqrt{5} \pm \sqrt{3})^2}{2} $


    Then [1] becomes: .$\displaystyle z^2 \;=\;\dfrac{(\sqrt{5}+\sqrt{3})^2}{2}$

    . . Hence: .$\displaystyle z \;=\;\pm\left(\dfrac{\sqrt{5}\pm\sqrt{3}}{\sqrt{2} }\right) $ . . . all real numbers.


    The locus is four vertical lines:

    . . $\displaystyle \begin{array}{ccccc}
    x &=& \dfrac{\sqrt{5}+\sqrt{3}}{\sqrt{2}} & \approx & 2.806\\ \\[-3mm]
    x &=& \dfrac{\sqrt{5} - \sqrt{3}}{\sqrt{2}} & \approx & 0.356\\ \\[-3mm]
    x &=& \dfrac{-\sqrt{5} + \sqrt{3}}{\sqrt{2}} & \approx & \text{-}0.356\\ \\[-3mm]
    x &=& \dfrac{-\sqrt{5} - \sqrt{3}}{\sqrt{2}} & \approx & \text{-}2.806 \end{array}$

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  4. #4
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    Ok thanks. Much easier than i had anticpiated.
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  5. #5
    Super Member Failure's Avatar
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    Quote Originally Posted by Soroban View Post
    The locus is four vertical lines:

    . . $\displaystyle \begin{array}{ccccc}
    x &=& \dfrac{\sqrt{5}+\sqrt{3}}{\sqrt{2}} & \approx & 2.806\\ \\[-3mm]
    x &=& \dfrac{\sqrt{5} - \sqrt{3}}{\sqrt{2}} & \approx & 0.356\\ \\[-3mm]
    x &=& \dfrac{-\sqrt{5} + \sqrt{3}}{\sqrt{2}} & \approx & \text{-}0.356\\ \\[-3mm]
    x &=& \dfrac{-\sqrt{5} - \sqrt{3}}{\sqrt{2}} & \approx & \text{-}2.806 \end{array}$
    No, I don't believe that that's quite right. Going back to $\displaystyle z^2=4\pm\sqrt{15}$ and setting $\displaystyle z := x+iy$, we get $\displaystyle (x+iy)^2=x^2+2ixy-y^2=4\pm\sqrt{15}$.
    Comparing real- and imaginary parts of the left and right hand side we see that we must have $\displaystyle xy=0$ and $\displaystyle x^2-y^2=4\pm\sqrt{15}$.

    But $\displaystyle x=0$ is not possible, because then the second of these two equations would not have a (real) solution y.

    Setting $\displaystyle y=0$ to satisfy the first equation gives the solutions $\displaystyle x_{1,2,3,4}=\pm\sqrt{4\pm\sqrt{15}}$ of the second equation.

    Hence I get $\displaystyle z_{1,2,3,4}=\pm\sqrt{4\pm\sqrt{15}}$ four isolated points (and not four vertical lines).
    Last edited by Failure; Aug 30th 2010 at 07:09 AM. Reason: fixed wrong signs in expansion of (x+iy)^2=
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  6. #6
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    So what is the locus?
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  7. #7
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    Quote Originally Posted by Lukybear View Post
    So what is the locus?
    As I wrote, it is just the four points $\displaystyle z_{1,2,3,4}=\pm\sqrt{4\pm \sqrt{15}}$ on the real axis. This agrees with the solution Soroban gave you, with the small difference that y cannot be arbitrary, but must be =0.

    If you take $\displaystyle z:= \sqrt{4+\sqrt{15}}+i\cdot 2$, for example, which would lie on one of those vertical axes Soroban found, you get that the left hand side of the original equation evaluates to $\displaystyle z^2+1/z^2\approx 3.90+i\cdot 11.14$, which is rather too far away from the value of the right hand side, $\displaystyle 8$.
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