Question:
The complex number $z = z + iy$ which satisfies the equation $\left| \frac{z-3i}{z+3i}\right| = 1 $, lies on
The complex number $z = z + iy$ which satisfies the equation $\left| \frac{z-3i}{z+3i}\right| = 1 $, lies on
Updated On: Jan 30, 2025
- the X-axis
- the straight line y = 3
- a circle passing through origin
- None of the above
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The Correct Option is A
Solution and Explanation
$\left| \frac{z-3i}{z+3i}\right| = 1 $
$ \Rightarrow \left|z-3i\right| = \left|z+3i\right| $
[if $| z-z_1 |=| z + z_2 |$ ,
then it is a perpendicular bisector of $z_1$ and $z_2$]
Hence, perpendicular bisector of $(0, 3)$ and $(0, - 3)$ is X-axis.
$ \Rightarrow \left|z-3i\right| = \left|z+3i\right| $
[if $| z-z_1 |=| z + z_2 |$ ,
then it is a perpendicular bisector of $z_1$ and $z_2$]
Hence, perpendicular bisector of $(0, 3)$ and $(0, - 3)$ is X-axis.
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