The minimum value of the expression E = z 2 + z - 3 2 + z - 6 i 2 (where z is a complex number) is equal to

Step-by-step solution

Let z = x + i y E = z z ¯ + z − 3 z ¯ − 3 + z − 6 i z ¯ + 6 i = 3 z z ¯ − 3 z + z ¯ + 9 + 6 z − z ¯ i + 36 z + z ¯ = 2 x ; z − z ¯ = 2 i y = 3 x 2 + y 2 - 6 x - 12 y + 45 = 3 x 2 + y 2 - 2 x - 4 y + 15 E = 3 x - 1 2 + y - 2 2 + 10 Hence, E m i n = 30 when x = 1 and y = 2 i.e., z = 1 + 2 i