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Thread: Complex Numbers - Circles

  1. #1
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    Complex Numbers - Circles

    Determine the set of solutions for $\displaystyle M = z \epsilon C | |z-3| = 2|z+3| $

    Hint: Try to find an equation in form of a general Circle.

    Seriously i have no idea.. im quite good in complex numbers, but this is somehow hard for me
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  2. #2
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    Hello, coobe!

    Determine the set of solutions for: .$\displaystyle M \;=\;\bigg\{z \in C\;\bigg|\;|z-3| \,=\, 2|z+3|\,\bigg\} $

    Hint: Try to find an equation in form of a general circle.
    Let $\displaystyle P(x,y)$ be a solution to the equation.


    $\displaystyle |z-3|$ is the distance of point $\displaystyle P$ from point $\displaystyle A(3,0).$

    $\displaystyle |z+3|$ is the distance of point $\displaystyle P$ from point $\displaystyle B(-3,0). $


    We are told that: .$\displaystyle d(PA) \;=\;2\cdot d(PB)$

    . . Hence: .$\displaystyle \sqrt{(x-3)^2 + y^2} \;=\;2\sqrt{(x+3)^2 + y^2}$


    Square and expand: .$\displaystyle x^2-6x+9 + y^2 \;=\;4(x^2+6x+9 + y^2) $

    Simplify: .$\displaystyle 3x^2 + 30x + 3y^2 -27 \:=\:0 \quad\Rightarrow\quad x^2 + 10x + y^2 \:=\:-9 $

    Complete the square: .$\displaystyle x^2 + 10x \:{\color{red}+\: 25} + y^2 \;=\;-9 \:{\color{red}+\: 25 } $


    Therefore: .$\displaystyle (x+5)^2 + y^2 \:=\:16$

    . . The solution is a circle with center $\displaystyle (-5,0)$ and radius 4.


    Note: It is known as the Circle of Apollonius.
    .
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