Question

Let the minimum value v 0 of
v = |z|2+|z-3|2+|z-6i|2,z∈C
is attained at z = z 0. Then
$|2z^2_0-\overline{z}^3_0+3|^2+v^2_0$
is equal to

• A. 1000
• B. 1024
• C. 1105
• D. 1196

Answer 1

Answer: (A)

1000

Answer 2

The correct answer is (A):
Let z = x + _iy
_ 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(1 + 2 i)2 - (1 - 2 i)3 + 3|2 + 900
= |2(-3 + 4 i) - (1 - 8 i 3 - 6 i (1 - 2 i) +3|2 + 900
= |-6 + 8 i - (1 + 8 i - 6 i - 12) + 3|2 + 900
= |8 + 6 i |2 + 900
= 1000