Given Information:
- The equation of the circle is f(x,y) = 0.
- The equation f(0,k) = 0 has equal roots k = 1, 1.
- The equation f(k,0) = 0 has roots k = 1/5, 5.

Explanation:

Finding the Equation of the Circle:
To find the equation of the circle, we need to consider the general equation of a circle: (x-a)^2 + (y-b)^2 = r^2, where (a, b) is the center of the circle and r is the radius.

Finding the Center of the Circle:
To find the center of the circle, we can use the given information.
- The equation f(0,k) = 0 has equal roots k = 1, 1. This means that substituting x = 0 and y = 1 in the equation f(x,y) = 0 gives us two identical roots. Since this equation represents a circle, the x-coordinate of the center of the circle is 0.
- Similarly, the equation f(k,0) = 0 has roots k = 1/5, 5. Substituting x = 1/5 and y = 0 in the equation f(x,y) = 0 gives us one root, and substituting x = 5 and y = 0 gives us another root. Since this equation represents a circle, the y-coordinate of the center of the circle is 0.

Therefore, the center of the circle is (0, 0).

Finding the Radius of the Circle:
Now that we know the center of the circle, we can find the radius by substituting the coordinates of one of the given points on the circle into the equation of the circle.

Let's consider the point (0, 1), which satisfies the equation f(0,k) = 0 with k = 1.
Substituting x = 0 and y = 1 in the equation f(x,y) = 0, we get f(0,1) = 0, which gives us the squared radius as (0-0)^2 + (1-0)^2 = 1.

Therefore, the radius of the circle is 1.

Summary:
- The equation of the circle is f(x,y) = 0.
- The center of the circle is (0, 0).
- The radius of the circle is 1.