Bordered Hessian. I have no idea.

Question

On plane $OXY$ find point $P(x,y)$, for which sum of squared distances to given points $P_{i}(x_i,y_i)$, $i=1,2,...,n$ is the smallest possible.

Answer 1

Assuming that the $(x,y)$ are cartesian coordinates (your minimalist question and useless title are unclear), this is a simple optimization problem. You are looking for the values of $x$ and $y$ which minimize the sum $$\sum_{i=1}^n (x-x_i)^2 + (y-y_i)^2$$ Note that we can break the sum in two with the first half depending only on $x$ and the second half depending only on $y$. If we minimize the two halves separately, it will minimize the whole as well.

So it is just a matter of finding the value of $x$ which minimizes $f(x) =\sum_{i=1}^n (x-x_i)^2$. Setting the derivative to zero, the minimum will occur when $$2\sum_{i=1}^n (x-x_i) = 0$$ or $$x =\frac {\sum_{i=1}^n x_i}n$$ I.e., when $x$ is the average of $x_i$. Similarly the second half is minimized when $y$ is the average of the $y_i$:

$$(x,y) =\left(\frac {\sum_{i=1}^n x_i}n,\frac {\sum_{i=1}^n y_i}n\right)$$