If () is a purely imaginary number and , then a value of is
- A.
- B.
- C.
- D.
Show answer and solution
Correct option: D
Solution
Let z = x + iy. Since (z-α)/(z+α) is purely imaginary, its real part is zero.
Multiplying numerator and denominator by the conjugate of (z+α):
Re[(z-α)(z̄+α)/|z+α|²] = 0
(z-α)(z̄+α) = |z|² + αz - αz̄ - α² = |z|² - α² + α(z - z̄)
The term α(z - z̄) = 2αi·y is purely imaginary, so Re part = |z|² - α².
Setting this to zero: |z|² = α², so α = |z| = 2.
Multiplying numerator and denominator by the conjugate of (z+α):
Re[(z-α)(z̄+α)/|z+α|²] = 0
(z-α)(z̄+α) = |z|² + αz - αz̄ - α² = |z|² - α² + α(z - z̄)
The term α(z - z̄) = 2αi·y is purely imaginary, so Re part = |z|² - α².
Setting this to zero: |z|² = α², so α = |z| = 2.
[JEE Main 2019 · 12 Jan · Shift I]