Answer: the equation of the circle in standard form is (x + 3)^2 + (y + 11)^2 = 121.
Step-by-step explanation: To find the center of the circle, we can substitute the coordinates of the point where the circle is tangent to the x-axis, which is (-3,0), into the equation of the line y = 4x + 1.
Substituting -3 for x, we get:
y = 4(-3) + 1
y = -12 + 1
y = -11
Therefore, the center of the circle is (-3, -11).
To find the radius, we need to measure the distance from the center of the circle to any point on the circle. Since the circle is tangent to the x-axis at (-3,0), the distance from the center to this point is the radius.
Using the distance formula, we can calculate the radius:
r = √[(x2 - x1)^2 + (y2 - y1)^2]
r = √[(-3 - (-3))^2 + (0 - (-11))^2]
r = √[(0)^2 + (11)^2]
r = √(0 + 121)
r = √121
r = 11
Therefore, the radius of the circle is 11.
Now we have the center (-3, -11) and the radius 11. We can plug these values into the standard equation of a circle to get the equation in standard form:
(x - h)^2 + (y - k)^2 = r^2
(x - (-3))^2 + (y - (-11))^2 = 11^2
(x + 3)^2 + (y + 11)^2 = 121