circle arc through two points, orthogonal to a given circle

I'm not sure to understand exactly the question but perhaps you want something like this.

documentclass{standalone}
usepackage{tkz-euclide}

begin{document}
begin{tikzpicture}[scale=3]
  tkzDefPoint(0,0){O}
  tkzDefPoint(1,0){A}
  tkzDrawCircle[blue](O,A)    
  tkzDefPoint(-.5,-.5){z1}
  tkzDefPoint(.5,-.25){z2} 
  tkzDefCircle[orthogonal through=z1 and z2](O,A) tkzGetPoint{B}
  tkzInterCC(O,A)(B,z1) tkzGetPoints{C}{D}  
  tkzDrawPoints[size=4](O,z1,z2,C,D,B) 
  tkzClipCircle(O,A)
  tkzDrawCircle[thick,color=red](B,z1)
end{tikzpicture} 

end{document} 

You can adapt this code with only TikZ

You mean something like this?

documentclass[tikz,border=5pt]{standalone}
usetikzlibrary{calc}

begin{document}

begin{tikzpicture}
path (0.6,0.8) coordinate (p) (-0.8,0.6) coordinate (q);
node at (p) [fill,circle,inner sep=0.7pt,label=$p$]{} node at (q) [fill,circle,inner sep=0.7pt,label=$q$]{};
path ($(p)!0.5!(q)$) coordinate (c) ($(0,0)!2!(c)$) coordinate (Y);

draw [clip] circle(1);
draw [blue] (Y) circle(1);
end{tikzpicture}

end{document}