Tikz: path generating a large blank space

Let’s suppose the following situation (wich is the simplest I could think): a circle intersepts two different points of a line and we call these points C and D.

documentclass[a4paper, 10pt]{article}
    usepackage{tikz}
    usetikzlibrary{intersections}
begin{document}
And here we've got a text about whatever it's talking about and probably is introducing this:
    
    begin{figure}[h!]
    centering
        begin{tikzpicture}
        coordinate (A) at (-2,0);
        coordinate (B) at (2,0);
        coordinate (P) at (0,1);
        draw [name path=line] (A)--(B);
        draw [name path=circle] (P) circle (1.5cm);
        path [name intersections ={of=line and circle,name=N,total=t}];
        node at (N-1)[below left]{C};
        node at (N-2)[below right]{D};
        end{tikzpicture}
    end{figure}

Which lead us to the meaning of the figure.
end{document}

My situation actually uses the through library, would be, in the example, something like giving a point (C or D) of the line and drawing a circle from P passing through this point. I don’t really know if it does change the problem. The thing is: what if I just wanted the coordinates? Instead of using draw, I’d use path (or node at (?) [circle through (…)];) and whatever commands to simply get the coordinates. In the example, I just switched draw [name path=circle] (P) circle (1.5cm); to path [name path=circle] (P) circle (1.5cm);

documentclass[a4paper, 10pt]{article}
    usepackage{tikz}
    usetikzlibrary{intersections}
begin{document}
And here we've got a text about whatever it's talking about and probably is introducing this:
    
    begin{figure}[h!]
    centering
        begin{tikzpicture}
        coordinate (A) at (-2,0);
        coordinate (B) at (2,0);
        coordinate (P) at (0,1);
        draw [name path=line] (A)--(B);
        path [name path=circle] (P) circle (1.5cm);
        path [name intersections ={of=line and circle,name=N,total=t}];
        node at (N-1)[below left]{C};
        node at (N-2)[below right]{D};
        end{tikzpicture}
    end{figure}

Which lead us to the meaning of the figure.
end{document}

This huge blank space. How to get rid of it in order to get something like this:

By the way, for the last image I just typed C and D coordinates as approximations of the actual X coordinate for both of them.

documentclass[a4paper, 10pt]{article}
usepackage{tikz}
usetikzlibrary{intersections}
begin{document}
And here we've got a text about whatever it's talking about and probably is introducing this:
    
    begin{figure}[h!]
    centering
        begin{tikzpicture}
        path (-2,0) coordinate (A) (2,0) coordinate (B);
        path[overlay] (0,1) coordinate (P);
        draw [name path=line] (A)--(B);
        path[overlay,name path=circle] (P) circle[radius=1.5cm];
        path [name intersections ={of=line and circle,name=N}];
        node at (N-1)[below left]{C};
        node at (N-2)[below right]{D};
        end{tikzpicture}
    end{figure}

Which lead us to the meaning of the figure.
end{document}

documentclass[a4paper, 10pt]{article}
    usepackage{tikz}
    usetikzlibrary{intersections,fit}
begin{document}
And here we've got a text about whatever it's talking about and probably is introducing this:
    
    begin{figure}[h!]
    centering
        begin{tikzpicture}
        coordinate (A) at (-2,0);
        coordinate (B) at (2,0);
        coordinate (P) at (0,1);
        draw [name path=line] (A)--(B);
        path [name path=circle] (P) circle (1.5cm);
        path [name intersections ={of=line and circle,name=N,total=t}];
        node (C) at (N-1)[below left]{C};
        node (D) at (N-2)[below right]{D};
        pgfresetboundingbox
        node[fit=(A) (B) (C) (D)] {};
        draw[red] (current bounding box.south west) rectangle (current bounding box.north east);
        end{tikzpicture}
    end{figure}

Which lead us to the meaning of the figure.
end{document}