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  4. All complex numbers z which satisfy the equation |(z-6i/z+6i)| = 1 lie on the

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Q. All complex numbers z which satisfy the equation $\left|\frac{z-6i}{z+6i}\right| = 1$ lie on the

COMEDKCOMEDK 2009Complex Numbers and Quadratic Equations

Solution:

Let $z =x + iy $
$\therefore \:\: \left|\frac{x+iy -6i}{x+ iy + 6i}\right| = 1$ or $ \frac{\left|x+i\left(y-6\right)\right|}{\left|x+i\left(y+6\right)\right|} = 1 $
$\Rightarrow \sqrt{x^{2} +\left(y -6\right)^{2} } = \sqrt{x^{2} + \left(y +6\right)^{2}} $
$\Rightarrow x^{2} + y^{2} +36 - 12y = x^{2} +y^{2} + 36 + 12 y $
$\Rightarrow 24y =0 \Rightarrow y = 0$
$ \therefore \:\:\: z = x $ i.e., $z$ lies on the real axis.