Algebra
Complex Numbers
MMTS_Full_Test_16
Grade 12
Question:
For any complex number $z$, $|z|^2+|z-1|^2+|z-2|^2+|z-i|^2$ is minimum when $z=$
$\dfrac{3}{4}+\dfrac{i}{4}$
$\dfrac{1}{2}+\dfrac{i}{4}$
$1+i$
$0$
Step-by-Step Solution
Key Concept: Minimize $f(z)=|z-z_k|^2$ over $k$; minimum at centroid of $0,1,2,i$
$z=\frac{0+1+2+i}{4}=\frac{3+i}{4}=\frac{3}{4}+\frac{i}{4}$.
Correct Answer: 1