Solveeit Logo
Login

Question

Question: $z_1=1+i$ $z_2=-2+3i$ $z_3=3-4i$ A Point A(z) moves on the plane such that $|z-z_1|^2+|z-z_2|^2+|z-z...

z1=1+iz_1=1+i z2=−2+3iz_2=-2+3i z3=3−4iz_3=3-4i A Point A(z) moves on the plane such that ∣z−z1∣2+∣z−z2∣2+∣z−z3∣2|z-z_1|^2+|z-z_2|^2+|z-z_3|^2 is minimum, then z

A

2/3

B

1+i

C

-2+3i

D

3-4i

Answer

2/3

Explanation

Solution

The expression ∣z−z1∣2+∣z−z2∣2+∣z−z3∣2|z-z_1|^2+|z-z_2|^2+|z-z_3|^2 represents the sum of the squared distances from a point zz to the points z1,z2,z3z_1, z_2, z_3 in the complex plane. This sum is minimized when zz is the geometric centroid of the points z1,z2,z3z_1, z_2, z_3. The centroid is calculated as the average of the complex numbers: z=z1+z2+z33z = \frac{z_1+z_2+z_3}{3}.

Given z1=1+iz_1 = 1+i, z2=−2+3iz_2 = -2+3i, and z3=3−4iz_3 = 3-4i: z1+z2+z3=(1+i)+(−2+3i)+(3−4i)z_1+z_2+z_3 = (1+i) + (-2+3i) + (3-4i) z1+z2+z3=(1−2+3)+(1+3−4)iz_1+z_2+z_3 = (1 - 2 + 3) + (1 + 3 - 4)i z1+z2+z3=2+0i=2z_1+z_2+z_3 = 2 + 0i = 2

Therefore, z=23z = \frac{2}{3}.