JEE AdvancedHardQuestion 230020
Question
If |z| = 1 and z ≠ ±1, then all values of z/(1 − z²) lie on:
- AImaginary axisCorrect
- BReal axis
- CUnit circle
- DParabola
Correct answer
Imaginary axis
Explanation
Let z = cos θ + i sin θ (|z|=1). Then 1 − z² = 1 − cos 2θ − i sin 2θ = 2sin²θ − 2i sinθ cosθ = 2sinθ(sinθ − i cosθ). And z/(1−z²) = (cosθ + i sinθ)/(2sinθ(sinθ − i cosθ)). Numerator × conjugate of denominator divided by |denominator|²: this yields a purely imaginary number. So values lie on the imaginary axis.