Question

The least value of $|z-3-4i{|}^2+|z+2-7i{|}^2+|z-5+2i{|}^2$ occurs when $z=x+iy$ then the real part of $z$ is

• A. $2$
• B. $3$
• C. $\sqrt{1}3$
• D. None of these

Answer 1

The correct option is A $2$

Let $z=x+iy$ then
$|z-3-4i{|}^2+|z+2-7i{|}^2+|z-5+2i{|}^2=|(x-3)+(y-4)i{|}^2+|(x+2)+(y-7)i{|}^2+|(x-5)+(y+2)i{|}^2=3(x^2+y^2-4x-6y)+107=3\left[\right(x-2{)}^2+(y-3{)}^2]+68$
This will have least value if $x=2\&y=3$
So, real part of $z=2$