Let the minimum value v 0 of v = z 2 + z - 3 2 + z - 6 i 2 , z ∈ ℂ is attained at z = z 0 . Then 2 z 0 2 - z ¯ 0 3 + 3 2 + v 0 2 is equal to

Step-by-step solution

Given, the minimum value v 0 of v = z 2 + z - 3 2 + z - 6 i 2 , z ∈ ℂ is attained at z = z 0 , Let z = x + i y v = x 2 + y 2 + x - 3 2 + y 2 + x 2 + y - 6 2 = 3 x 2 - 6 x + 9 + 3 y 2 - 12 y + 36 = 3 x 2 + y 2 - 2 x - 4 y + 15 = 3 x - 1 2 + y - 2 2 + 10 V min at z = 1 + 2 i = z 0 and v 0 = 30 So, 2 z 0 2 - z ¯ 0 3 + 3 2 + v 0 2 = 2 1 + 2 i 2 - 1 - 2 i 3 + 3 2 + 900 = - 6 + 8 i - 1 + 8 i - 6 i - 12 + 3 2 + 900 = 8 + 6 i 2 + 900 = 1000