All circles can be written as x^2 + y^2 = r^2.

This is a circle centered about the origin (0,0) with radius r.

A circle of at (3,-1) would be (x-3)^2 + (y+1)^2 = r^2

The same circle with diameter 2 would be (x-3)^2 + (y+1)^2 = 4

In your problem, you should draw out the circle.

There are two possible circles.

One that is below the x-axis, but touches at (4,0), and one that is above the x-axis, but touches at (4,0).

You have a radius of 5, so for one circle, go up 5 in the y-direction, and for the other, go down 5 in the y-direction.

The centers (a,b) would be (4,5) and (4,-5).

You know how to write these now:

A circle at (4,5) would be (x-4)^2 + (y-5)^2 = r^2

A circle at (4,-5) would be (x-4)^2 + (y+5)^2 = r^2

And you know the radius is 5, so r^2 equals (5^2) equals 25.

A circle at (4,5) with radius 5 would be (x-4)^2 + (y-5)^2 = 25

A circle at (4,-5) with radius 5 would be (x-4)^2 + (y+5)^2 = 25

^Cartesian equations.