Question 990645
Since the circle touches both the axes,
we, can take center of circle as (r,r) where r is also radius of circle.
(Draw a fig. to see clearly)
So, equation of circle can be given by,
{{{(x-r)^2 + (y-r)^2 = r^2}}} --------------(i)
Since, it passes through (1,2),
{{{(1-r)^2 + (2-r)^2 = r^2}}}
=> {{{ r^2-2r+1 + r^2-4r+4 = r^2}}}
=> {{{r^2-6r+5 = 0}}}
=> {{{(r-1)(r-5) = 0}}}
So r = 1 or r =5
So, clearly we have two circles with centres at (1,1) and (5,5) with radii 1 and 5 respectively.
Equation of circles be given by putting value of r in eqn (i):- 
{{{(x-1)^2 + (y-1)^2 = 1^2}}}
and
{{{(x-5)^2 + (y-5)^2 = 5^2}}}
(Simplify it.)

A good coordinate geometry book:- "The Elements of Coordinate geometry" by S. L. Loney