If and , then implies that, in the complex plane
- A.lies on the imaginary axis
- B.lies on the real axis
- C.lies on the unit circle
- D.None of these
Show answer and solution
Correct option: B
Solution
Substitute z = x + iy into w = (1 - iz)/(z - i).
1 - iz = 1 - i(x + iy) = 1 + y - ix
z - i = x + i(y - 1)
|w| = 1 means |1 - iz| = |z - i|, so:
(1 + y)² + x² = x² + (y - 1)²
1 + 2y + y² = y² - 2y + 1
4y = 0, so y = 0.
Since y = Im(z) = 0, z lies on the real axis.
1 - iz = 1 - i(x + iy) = 1 + y - ix
z - i = x + i(y - 1)
|w| = 1 means |1 - iz| = |z - i|, so:
(1 + y)² + x² = x² + (y - 1)²
1 + 2y + y² = y² - 2y + 1
4y = 0, so y = 0.
Since y = Im(z) = 0, z lies on the real axis.
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