Complex NumbersHard
Question
If z = x + iy and w = (1 - iz) / (z - i), then |w| = 1 implies that, in the complex plane
Options
A.z lies on the imaginary axis
B.z lies on the real axis
C.z lies on the unit circle
D.None of the above
Solution
Since, |w| = 1
⇒
= 1 ⇒ |z - i| = |1 - iz|Since, |w| = 1
⇒ = 1 ⇒ |z - i| = |1 - iz|
⇒ |z - i|=|z + i|
(∵ |1 - iz |=|- i||z + i|=|z + i|)
∴ It is a perpendicular bisector of (0,1) and (0, - 1) ie, x-axis
Thus, z lies on real axis.
⇒
= 1 ⇒ |z - i| = |1 - iz|Since, |w| = 1⇒
= 1 ⇒ |z - i| = |1 - iz|⇒ |z - i|=|z + i|
(∵ |1 - iz |=|- i||z + i|=|z + i|)
∴ It is a perpendicular bisector of (0,1) and (0, - 1) ie, x-axis
Thus, z lies on real axis.
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