A point z moves in the complex plane such that arg z - 2 z + 2 = π 4 , then the minimum value of | z - 9 2 - 2 i | 2 is equal to

Step-by-step solution

Given that arg z - 2 z + 2 = π 4 Let z   =   x   +   i y ⇒ arg x + i y - 2 x + i y + 2 = π 4 ⇒ arg x + i y - 2 x + i y + 2 x + 2 - i y x + 2 - i y = π 4 ⇒ arg x 2 + y 2 - 4 + 4 i y ( x + 2 ) 2 + y 2 = π 4 ⇒ x 2 + y 2 - 4 + 4 i y ( x + 2 ) 2 + y 2 = tan π 4 ⇒ x 2 + y 2 - 4 4 y = 1 x 2 + ( y - 2 ) 2 = 8 ……….(1) Equation ( 1 ) represents circle, whose centre ( 0 , 2 ) , radius 2 2 . Minimum distance d = | A   C - r | G