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The complex number $z$, satisfying the equation $|\large\frac{z-5i}{z+5i}|=$$1$ lies on?

(A) A line $ y=5$ (B) A circle through origin (C) $x-axis$ (D) $y-axis$
Can you answer this question?
 
 

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  • Equation of $x-axis$ is $y=0$
Let $z=x+iy$
$|\large\frac{z-5i}{z+5i}|=|\frac{x+iy-5i}{x+iy+5i}|$$=1$
$\Rightarrow\:|\large\frac{\sqrt {x^2+(y-5)^2}}{\sqrt {x^2+(y+5)^2}}|$$=1$
$\Rightarrow\:x^2+(y-5)^2=x^2+(y+5)^2$
$\Rightarrow\:(y-5)^2=(y+5)^2$
$\Rightarrow\:-10y=10y$
$\Rightarrow\:y=0$
$\therefore z$ lies on $x-axis$
answered Jul 4, 2013 by rvidyagovindarajan_1
 

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