If a complex number z is such that ( 7 + i ) ( z + z ¯ ) - ( 4 + i ) ( z - z ¯ ) + 116 i = 0 , then z z ¯ =

Step-by-step solution

Consider a complex equation, z = x + i   y Its conjugate is z ¯ = x - i y Now, ( 7 + i ) ( z + z ¯ ) - ( 4 + i ) ( z - z ¯ ) + 116 i = 0 ⇒ ( 7 + i ) ( 2 x ) - ( 8 i y - 2 y ) + 116 i = 0 ⇒ ( 14 x + 2 y ) + i ( 2 x - 8 y + 116 ) = 0 The equations obtained are ⇒ 2 x - 8 y + 116 = 0 ⇒ 14 x + 2 y = 0 From the above two equations, x = - 2 ,   y = 14 The modulus is | z | = x 2 + y 2 = ( - 2 ) 2 + 14 2 = 200 Thus, z z ¯ = | z | 2 = ( 200 ) 2 = 200