Given $n$ points $P_{1},P_{2},\dots P_{n}$ and a real number $c,$ find the locus of points $X$ such that $$\sum_{i=1}^{n}XP_{1}^{2}=c.$$
Actually, I'm also interested in a more general case: Given $n$ points $P_{1},P_{2},\dots ,P_{n},$ and reals $r_{1},r_{2},\dots r_{n},$ and $c,$ find the locus of points $X$ such that $$\sum_{i=1}^{n}r_{1}XP_{1}^{2}=c.$$
Is the locus anything significant? Is there a way of constructing it?
Also, is there a way of finding this locus without resorting to analytical techniques?
Thanks.
@Aretino has answered you concerning an analytical technique.
Here is a (classical) vector technique for this issue.
Consider the case of three points $P_1,P_2,P_3$ with centroid $G$ (the general case with $n$ points and hopefully weights for these points is completely similar).
We are looking for the locus of points $X$ such that
$$\varphi(X)=(XP_1)^2 + (XP_2)^2 + (XP_3)^2=c, \ \ \text{a given constant}$$
But we can write :
$$(XP_1)^2 + (XP_2)^2 + (XP_3)^2=\overrightarrow{XP_1}^2+\overrightarrow{XP_2}^2+\overrightarrow{XP_3}^2=$$
$$=(\overrightarrow{XG}+\overrightarrow{GP_1})^2+(\overrightarrow{XG}+\overrightarrow{GP_2})^2+(\overrightarrow{XG}+\overrightarrow{GP_3})^2$$
Expanding, we get :
$$3 \overrightarrow{XG}^2+2\overrightarrow{XG} . \underbrace{\sum_i \overrightarrow{GP_i}}_{= \ 0}+\underbrace{\sum_i \overrightarrow{GP_i}^2}_{\varphi(G) = \text{constant} \ k} = c.$$
Therefore, if $c \ge k$, the locus of points $M$ is a circle centered in $G$ with radius $\sqrt{\frac{c-k}{3}}$.