Radical axis generalization I’ll denote power of point $X$ respect circle $\omega$ by $\mathcal{P}_{\omega}(X)$.
Radical axis of circles $\omega_1, \omega_2$ is locus of points satisfying the following condition:
$$\mathcal{P}_{\omega_1}(X)= \mathcal{P}_{\omega_2}(X)$$
I know that for $k\in\mathbb{R}$, locus of points satisfying $$\mathcal{P}_{\omega_1}(X)=k\cdot \mathcal{P}_{\omega_2}(X)$$ is line or circle. Let this circle be $\omega_0$. And I proved that circles $\omega_0, \omega_1, \omega_2$ are coaxial.
Is there any name of circle $\omega_0$? And any materials related to this? I really want to learn about this.
 A: Since this is in part a reference request, I'd recommend the paper Pfiefer and van Hook, Circles, Vectors, and Linear Algebra.
(In the following I'm going to abuse notation and switch freely between $\omega, c, c(x,y),$ and $c(x,y)=0$ as ways of designating circles.)
This paper treats circles $c$ via their equations $c(x,y)=0$, where
$$
c(x,y)=\alpha(x^2+y^2)-2\beta x^2-2\gamma x^2+\delta.
$$
and views them through the lens of linear algebra (vectors, dot products, linear combinations, etc).
How does this relate to the power of a point $\mathcal{P}_{\omega}(X)?$
It turns out that  $\mathcal{P}_{c}((x,y))=c(x,y).$  In other words, the power of point $(x,y)$ wrt to the circle $c(x,y)=0$ is $c(x,y)$.
So the circle corresponding to the locus of points satisfying
$$
\mathcal{P}_{\omega_1}(X)=k\cdot \mathcal{P}_{\omega_2}(X)
$$
is the circle
$$
c_0(x,y)=c_1(x,y)-k\cdot c_2(x,y)=0.
$$
In general, the circles $c_0$ don't have a name.  But you've already spotted the special case $k=1$, which gives the radical axis.  And if $r_1,r_2$ are the circle radii, then $k=\pm r_1/r_2$ give the circles of antisimilitude, which invert $c_1$ to $c_2$ and vice versa (see pgs 78,79 of the referenced article).
