Question

For a complex number $z,$ find the minimum value of \[|z - 3|^2 + |z - 5 + 2i|^2 + |z - 1 + i|^2.\]

Answer 1

`[|z - 3|^2 + |z - 5 + 2i|^2 + |z - 1 + i|^2]`

Let , complex number z be , z = x + iy .

Putting z in above equation , we get :

`=(x-3)^2+y^2+(x-5)^2+(y+2)^2+(x-1)^2+(y+1)^2`

Now , getting critical points by :

`f'(x) = 2(x-3)+2(x-5)+2(x-1) = 0\\\\3x-9=0\\\\x=3`

Also ,

`f'(y)=2y+2(y+2)+2(y+1)=0\\\\3y+3=0\\\\y=-1`

So , at point ( 3, -1 ) complex number given expression have minimum value.

`=(3-3)^2+(-1)^2+(3-5)^2+(-1+2)^2+(3-1)^2+(-1+1)^2\\\\=10`

Therefore, minimum value is 10.

Hence, this is the required solution.