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  4. For a complex number Z , the equation of the line of common chord of the circles |Z - 3|=2 and |Z|=2 is

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Q. For a complex number $Z$ , the equation of the line of common chord of the circles $\left|Z - 3\right|=2$ and $\left|Z\right|=2$ is

NTA AbhyasNTA Abhyas 2020Complex Numbers and Quadratic Equations

Solution:

$\left|Z - 3\right|=2\Rightarrow \left(Z - 3\right)\left(\bar{Z} - \bar{3}\right)=4$
$\Rightarrow Z\bar{Z}-3Z-3\bar{Z}+9=4...,\left(1\right)$
$\left|Z\right|=2\Rightarrow Z\bar{Z}=4.... \left(2\right)$
For the equation of common chord, Subtracting $\left(2\right)$ from $\left(1\right)$ , we get,
$-3Z-3\bar{Z}+9=0$
$\Rightarrow Z+\bar{Z}=3$