Find the radius of each circle that passes through (9,2), and is tangent to both x-axis and y-axis.

Draw a picture.  The center of the circle is at (h,k), and (9,2) lies on the circle.  
Since the circle is tangent to both the x- and the y-axes, h=r and k=r.

The distance from the center (r,r) to the point (9,2) is

d = r = sqrt( (r-9)^2 + (r-2)^2 ), or r^2 = (r-9)^2 + (r-2)^2
                                                 or r^2 = r^2 - 18r + 81 + r^2 - 4r + 4
Simpifying,                                    r^2 = 2r^2 - 22r + 85, or r^2 - 22r + 84 = 0.

This quadratic can be solved using the quadratic formula or some other approach.  The roots are 5 and 17.

Thus, the center of the circle is (5,5) and the radius is 5.

The equation of the circle is then (x-5)^2 + (y-5)^2 = 5^2.

Check:  does (9,2) satisfy this equation?  (9-5)^2 + (2-5)^2 = 25

                                                                      4^2 + 3^2 = 25 is TRUE.

You must now go thru the same procedure with r = 17.  Is

(x-9)^2 + (y-2)^2 = 25 when the center is (17,17)?