Distance from a fixed point to a circle

[Enrico Fermi]

There is a fixed point P(x_p,y_p), and a fixed circle C_(O,R), distance from P to variable point A(x,y) is r.Also there is a point K(x_k,y_k) that AK perpendicular to OP.
Then R^2-OK^2=r^2-KP^2
but KP=OP-OK;
R^2-OK^2=r^2-(OP-OK)^2
r^2-R^2=OP^2-2OP*OK
OK=OP/2+(R^2-r^2)/2OP
KP=OP/2-(R^2-r^2)/2OP
:r^2=R^2+(OP/2-(R^2-r^2)/2OP)^2+(OP/2+(R^2-r^2)/2OP)^2
According to wolfram alfa the solution is https://www.wolframalpha.cominput?i=r%5E2%3DR%5E2%2B(2d+--(R%5E2)%2Br%5E2)%2Fd)%5E2+--d%2B(-(R%5E2)%2Br%5E2)%2Fd)%5E2
Acording to me it was an equation in R^4+R^2, whatever be the solution what most shocks me is that r is suposed to vary on A, ((x-x_o)^2+(y-y_o)^2)) but in r^2=R^2+(OP/2-(R^2-r^2)/2OP)^2+(OP/2+(R^2-r^2)/2OP)^2 OP and R are both fixed as C_(OR) and P are fixed.
Is it me or is geometry not working?