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Given two points, the origin and (a,b) in R^2, I have convinced myself that there are four circles that pass through these points, two circles with larger radius so the arc is "shallower" and two with smaller radii with "longer" arcs joining the points. I am trying to calculate the equations of these circles. As it stands I think I have the equation of one:

I use (x-a)^2 + (y-b)^2 = r^2 and (x-0)^2 + (y-0)^2 = r^2

to get a^2 - 2ax = 2by-b^2

then sub in y = 0 to get x value.

I then use these two to get an expression for r^2.

Giving r^2 = (a^2-b^2)/2a

and hence

[x- (a^2-b^2)/2a]^2 + y^2 = (a^2-b^2)/2a

Can anyone let me know how to calculate the other circles, or suggest a strategy?

Maybe the two arcs of the smaller radii circles are the same circle...

I know that if I reflect the circle that I do have the equation for in the line joining (0,0) to(a,b) I will get the equation of one other, but what about the missing two? And which one have I got now?

Thanks