It's easiest to work with the equation of the circle given by
$$(x-a)^2 + (y-b)^2=r^2$$
as this is the circle with center $(a,b)$ and radius $r$.
You are given two facts:
- The circle touches the $x$ axis at $(4,0)$.
- The circle passes through the point $(2,2)$.
The first fact tells you that the center of the circle must have an $x$ coordinate of $4$ and that the radius of the circle is the same as the $y$-coordinate.
Your job: Try to put the two things into equations. Knowing that $a$ is the $x$-coordinate and $b$ the $y$-coordinate of the center, and that $r$ is the radius, what equation does the sentence "The $x$ coordinate of the center is $4$" describe? What about the sentence "The radius is the same as the $y$ coordinate"?
The second fact tells you that plugging in $x=2$ and $y=2$ into the circle equation will result in a true statement. This should give you the third equation (the first fact gives you two), and you will have $3$ equations for $3$ variables, $a,b$ and $r$.