Weird coordinates

$k=(x-x_1)^{2}+(y-y_1)^{2}+(x-x_2)^{2}+(y-y_2)^{2}......$

Differentiating by first keeping y constant and then x

$\frac{dk}{dx}=2x-2x_{1}+2x-2x_{2}+...+2x-2x_{n}$

and

$\frac{dk}{dy}=2y-2y_{1}+2y-2y_{2}.......+2y-2y_{n}$

equating

$\frac{dk}{dx}=0$ and $\frac{dk}{dy}=0$

we get $x=\frac{x_{1}+x_{2}+....x_{n}}{n}$

$y=\frac{y_{1}+y_{2}+....y_{n}}{n}$

Using second derivative test

$k$ is minimum for the values of $x$ and $y$ .

Thus, minimum value of $k$ is sum of squares of distances from $centroid$

(denoted as $G$ )

$G(\frac{1+2+3+4+5}{5},\frac{1^{2}+2^{2}+3^{2}+4^{2}+5^{2}}{5})$

$G(3,11)$

$k=2^{2}+10^{2}+1^{2}+7^{2}+0^{2}+2^{2}+1^{2}+5^{2}+2^{2}+14^{2}$

$\boxed {k=384}$

plz upvote and follow