If z 1 + z 2 2 = z 1 2 + z 2 2 , where z 1 and z 2 are two complex numbers, then

Step-by-step solution

Given that: z 1 + z 2 2 = z 1 2 + z 2 2 , where z 1 and z 2 are two complex numbers z 1 + z 2 2 = z 1 2 + z 2 2 + 2 z 1 z 2 cos ( x - y ) , where x   &   y are arguments of z 1   &   z 2 . Here, 2 z 1 z 2 cos ( x - y ) = 0 ⇒ cos ( x - y ) = 0   ⇒ x - y = ( 2 n - 1 ) π 2   where   n ∈ I Now, a r g ( z 1 z 2 ) = a r g ( z 1 ) - a r g ( z 2 ) = x - y = ( 2 n - 1 ) π 2   where   n ∈ I ⇒ z 1 z 2 is a purely imaginary.