Let z 1 = 3 , z 2 = 2 and z 1 + z 2 + z 3 = 3 + 4 i . If the real part of z 1 z 2 - + z 2 z 3 - + z 3 z 1 - is equal to 4 , then z 3 is equal to (where, i 2 = - 1 )

Step-by-step solution

| z 1 + z 2 + z 3 | 2 = 3 2 + 4 2 = 25 ⇒ z 1 + z 2 + z 3 z 1 - + z 2 - + z 3 - = 25 ⇒ | z 1 | 2 + | z 2 | 2 + | z 3 | 2 + z 1 z 2 - + z 2 z 3 - + z 3 z 1 - + z 1 - z 2 + z 2 - z 3 + z 3 - z 1 = 25 ⇒ 9 + 4 + z 3 2 + z 1 z 2 - + z 2 z 3 - + z 3 z 1 - + z 1 - z 2 + z 2 - z 3 + z 3 - z 1 = 25 ⇒ z 3 2 + 2 r e a l ( z 1 z 2 - + z 2 z 3 - + z 3 z 1 - ) = 12 ⇒ | z 3 | 2 = 4 ⇒ z 3 = 2