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Thread: complex number

  1. #1
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    complex number

    the centre of circle represented by mod(z+1)=2mod(z-1) is
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  2. #2
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    Hello, prasum!

    $\displaystyle \text{The centre of circle represented by: }\;|z+1| \:=\:2|z-1|$

    $\displaystyle |z+1|\:=\:|z - (\text{-}1)|$ is the distance from $\displaystyle \,z$ to (-1,0).

    $\displaystyle |z-1|$ is the distance from $\displaystyle \,z$ to (1,0).


    Code:
                    |     P
                    |     o
                    | *    *
                  * |       *
         A    *     |        * B
      - - o - - - - + - - - - o - -
         -1         |         1
                    |

    We want points $\displaystyle P(x,y)$ so that: .$\displaystyle \overline{PA} \;=\;2\!\cdot\!\overline{PB}$
    [This is known as the Circle of Apollonius.]


    We have: .$\displaystyle \sqrt{(x+1)^2 + y^2} \;=\;2\sqrt{(x-1)^2+y^2}$

    Square: .$\displaystyle (x+ 1)^2 + y^2 \;=\;4\left[(x-1)^2 + y^2\right]$

    which simplifies to: .$\displaystyle 3x^2 - 10x + 3y^2 \;=\;-3$

    . . . . . . . . . . . . . . . . $\displaystyle x^2 - \frac{10}{3}x + y^2 \;=\;-1$

    Complete the square: .$\displaystyle x^2 - \frac{10}{3}x + \left(\frac{5}{3}\right)^2 + y^2 \;=\;-1 + \left(\frac{5}{3}\right)^2 $

    . . . . . . . . . . . . . . . . . . . . . $\displaystyle \left(x - \frac{5}{3}\right)^2 + y^2 \;=\;\frac{16}{9}$


    Therefore, the circle has center $\displaystyle (\frac{5}{3},\:0)$ radius $\displaystyle \frac{4}{3}$
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  3. #3
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    thanks
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