Consider the functions $$f(z_1,z_2,\dots,z_n) = c_1+\sum_{k=1}^n t_k(z_k-a_k)^2$$ and $$ g(z_1,z_2,\dots,z_n) = c_2+\sum_{k=1}^n t_k(z_k-a_k)^2 $$ where $c_1,c_2,t_k, a_k$ are real numbers.
Now consider the equation $$ c1+\sum_{k=1}^n t_k(z_k-a_k)^2=c2+\sum_{k=1}^n t_k(z_k-b_k)^2 $$
where $c_1, c_2$ are some constants.
Why does this define an $(n-1)$ dimension hyperplane?
Specifically, I am confused by the fact that there are $z_k$ being squared (I am used to seeing hyperplanes defined by $c_1x_1 + c_2x_2+\dots c_nx_n= 0$).
Perhaps $ c1+\sum_{k=1}^n t_k(z_k-a_k)^2=c2+\sum_{k=1}^n t_k(z_k-b_k)^2 $ can be rewritten to be of the form $c_1x_1 + c_2x_2+\dots c_nx_n= 0$?
Edit: As a side question, $f(z_1,\dots,z_n)=0$ would not define a hyperplane, would it? (I am uncertain because I don't know if we can think of the $(z_k-a_k)^2$ as constants and then this as defining a hyperplane with the $t_k$ corresponding to the $x_k$ in $c_1x_1 + c_2x_2+\dots c_nx_n= 0$
perhaps $f(z_1,\dots,z_n)=0$ defines a hyperplane for a given set of points $(z_1,\dots,z_n)$? I.e. maybe we can think of $f(z_1,\dots,z_n)\vert_{Z_1,Z_2,\dots Z_n}=0$ as $g(t_1,\dots,t_n)=0$?)