Among the statements (S1) : The set { z ∈ C − { − i } : ∣ z ∣ = 1 and z + i z − i is purely real} contains exactly two elements, and (S2) : The set { z ∈ C − { − 1 } : ∣ z ∣ = 1 and z + 1 z − 1 is

Question

Among the statements (S1) : The set { z ∈ C − { − i } : ∣ z ∣ = 1 and z + i z − i ​ is purely real} contains exactly two elements, and (S2) : The set { z ∈ C − { − 1 } : ∣ z ∣ = 1 and z + 1 z − 1 ​ is purely imaginary contains infinitely many elements.

TestHub · Accepted Answer

​ S 1 ​ : ∣ z ∣ = 1 , z + i z − i ​ = z − i z + i ​ ⇒ ( z − i ) ( z − i ) = ( z + i ) ( z + i ) ∣ z ∣ 2 − i ( z + z ) − 1 = ∣ z ∣ 2 + i ( z + z ) − 1 i ( z + z ) = 0 z + z = 2 cos θ = 0 ⇒ cos θ = 0 z = 0 + 0 i , ∣ z ∣  = 1 S 1 ​ : z + 1 z − 1 ​ + z + 1 z − 1 ​ = 0 ( z − 1 ) ( z + 1 ) + ( z + 1 ) ( z − 1 ) = 0 ⇒ ∣ z ∣ 2 + ( z − z ) − 1 + ∣ z ∣ 2 + ( z − z ) − 1 = 0 ∣ z ∣ 2 = 1 ​

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