### Instruction: Circles with centers of $(2,2)$ and $(17,10)$ are both tangent to the $x$-axis. What is the distance between the closest points of the two circles?### Response: I want to find the distance between the closest points of the two circles, so I need to know the radii of the circles and the distance between their centers.
Since the circles are tangent to the $x$-axis, their radii must be equal to their $y$-coordinates.
So the radius of the circle with center $(2,2)$ is $2$ units, and the radius of the circle with center $(17,10)$ is $10$ units.
To find the distance between their centers, I can use the distance formula: $d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$.
Plugging in the coordinates of the centers, I get $d = \sqrt{(17-2)^2+(10-2)^2} = \sqrt{225+64} = \sqrt{289} = 17$ units.
Now, the distance between the closest points of the circles is equal to the distance between their centers minus the sum of their radii.