If z ≠ i be any complex number such that z - i z + i is a purely imaginary number, then, z + 1 z is

Question

If z ≠ i be any complex number such that z - i z + i is a purely imaginary number, then, z + 1 z is, any non-zero real number other than 1 ., a purely imaginary number., 0, any non-zero real number

MARKS App · Accepted Answer

Let z = x + i y z - i z + i is a purely imaginary number ⇒ x + i y - 1 x + i y + 1 × x - i y + 1 x - i y + 1 is a purely imaginary ⇒ x 2 + y 2 - 1 - i 2 x x 2 + y + 1 2 is purely imaginary ⇒ x 2 + y 2 - 1 = 0 ⇒ x 2 + y 2 = 1 … i z + 1 z = x + i y + 1 x + i y = x + i y + 1 x + i y × x - i y x - i y = x + i y + x - i y x 2 + y 2 = 2 x y ≠ ± 1 so x ≠ 0 (from i and since, z won’t be an imaginary number)

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