Katie C. answered • 5d

Experienced Tutor Scored 33 ACT Math

The formula for a circle with centered at (a,b) with a radius of r is:

r = √((x-a)^2+(y-b)^2)

It's given that the center is (7, -1), so we can plug in a = 7 and b = -1:

r = √((x-7)^2+(y+1)^2)

Now you only need to find the radius, (the distance from the center of the circle to every point on the circle).

A *tangent line* on a circle is a line that intersects the circle at only one point, called *the tangent point* and runs perpendicular to the line from the center of the circle to the tangent point.

Since the x-axis (horizontal) is a tangent line, the line running from the center (7, -1) to the tangent point is vertical, meaning they have the same x-value. So for the tangent point x=7, and since the tangent line is on the x-axis we know that y=0 (the x-axis is all points satisfying y=0). So the point (7,0) is a point on our circle.

Using the distance formula, we have that the distance from (7, -1) to (7,0), our radius, is

r = √((7-7)^2+(-1-0)^2)

r = √1

r = 1

So our equation is

1 = √((x-7)^2+(y+1)^2)

You could also have plugged the point (7,0) with x=7 y=0 to

r = √((x-7)^2+(y+1)^2)

but thinking about this geometrically in this case is a little faster.