Question
Mathematics Question on Complex Numbers and Quadratic Equations
The value of ∣z∣2+∣z−3∣2+∣z−i∣2 is minimum when z equals.
A
232i
B
45+3i
C
1+3i
D
13i
Answer
1+3i
Explanation
Solution
∣z∣2+∣z−3∣2+∣z−1∣2 =x2+y2+(x−3)2+y2+x2(y−1)2 =3x2+3y2−6x−2y+10 =3(x−1)2+3(y2−32y)+10−3 =3(x−1)2+3(y2−32)2+7−3 =3z−(1+3i)2+320 This is minimum if z=1+3i