Where does z lie, if | z-5i z+5i |=1 - Vidyadip

Question

Where does z lie, if $\Big|\frac{\text{z}-5\text{i}}{\text{z}+5\text{i}}\Big|=1$

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Answer

Given that, $\Big|\frac{\text{z}-5\text{i}}{\text{z}+5\text{i}}\Big|=1$
Let $\text{z}=\text{x}+\text{yi}$
$\therefore\ \Big|\frac{\text{x}+\text{yi}-5\text{i}}{\text{x}+\text{yi}+5\text{i}}\Big|=1$
$\Rightarrow\Big|\frac{\text{x}+(\text{y}-5)\text{i}}{\text{x}+(\text{y}+5)\text{i}}\Big|=1$
$\Rightarrow|\text{x}+(\text{y}-5)\text{i}|=|\text{x}+(\text{y}+5)\text{i}|$
$\Rightarrow\text{x}^2+(\text{y}-5)^2=\text{x}^2+(\text{y}+5)^2$
$\Rightarrow(\text{y}-5)^2=(\text{y}+5)^2$
$\Rightarrow\text{y}^2+25-10\text{y}=\text{y}^2+25+10\text{y}$
$\Rightarrow20\text{y}=0$
$\Rightarrow\text{y}=0$
Hence, z lies on x-axis i.e., real axis.

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