# Thread: Complex Numbers - Circles

1. ## 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

2. 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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