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

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Solution

$z = x + i y$
$| z - 3 - 4 i |^{2} + | z + 2 - 7 i |^{2} + | z - 5 + 2 i |^{2}$
$= 3 (x^{2} + y^{2} - 4 x - 6 y) + 107$
$= 3 {(x - 2)^{2} + (y - 3)^{2}} + 68$
So, least value when $x = 2$ and $y = 3$
$z = 2 + 3 i$

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