A) the axis of\[x\]
B) the straight line\[y=5\]
C) the circle passing through the origin
D) none of the above
Correct Answer: A
Solution :
Given,\[z=x+iy\]and \[\left| \frac{z-5i}{z+5i} \right|=1\] \[\because \] \[\left| \frac{{{z}_{1}}}{{{z}_{2}}} \right|=\frac{|{{z}_{1}}|}{|{{z}_{2}}|}\] \[\Rightarrow \] \[\frac{|x+iy-5i|}{|x+iy-5i|}=1\] \[\Rightarrow \] \[|x+iy-5i|=|x+iy+5i|\] \[\Rightarrow \] \[{{x}^{2}}+{{(y-5)}^{2}}={{x}^{2}}+{{(5+y)}^{2}}\] \[\Rightarrow \] \[-10y=10y\Rightarrow y=0\] Which is the equation of axis of\[x\].
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