Q49 / 125 · BITSAT 2014 · Mathematics

The complex number $z=z+i y$ which satisfies the equation $\left|\frac{z-3 i}{z+3 i}\right|=1,$ lies on

  • Correct answer
    A
    the $X$ -axis
  • B
    the straight line $y=3$
  • C
    a circle passing through origin
  • D
    None of the above

Solution

$\left|\frac{z-3 i}{z+3 i}\right|=1 \Rightarrow|z-3 i|=|z+3 i|$ [if $\left|z-z_{1}\right|=\left|z+z_{2}\right|,$ then it is a perpendicular bisector of $z_{1}$ and $z_{2}$ ] Hence, perpendicular bisector of (0,3) and (0,-3) is $X$ -axis.

Hence, the correct answer is option: 'A'.

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