As the circle touches the x-axis at (4, 0) and the y-axis at (0, 4), the x and y axes are tangents to the circle. For any circle if a line perpendicular to a tangent is drawn from the point of tangency it passes through the center of the circle.
If lines are drawn from the points (4, 0) and (0, 4) perpendicular to the x-axis and y-axis respectively they intersect at the point (4, 4). This gives the center of the circle as (4, 4). The radius of the circle is the distance of (4, 4) from either axis and is equal to 4.
Using the radius of the circle and the its center, the equation of the circle is (x - 4)^2 + (y - 4)^2 = 16
The equation of the circle that touches the x-axis at (4, 0) and the y-axis at (0, 4) is (x - 4)^2 + (y - 4)^2 = 16