Answer:
The equation of the circle in standard form is:
(x + 1)² + (y - 2)² = 5
Explanation:
The center of the circle lies on the line y = 3x + 1.
Since the center is tangent to the x-axis at (−2, 0), this means the y-coordinate of the center is 0, and the x-coordinate is -2. So, the center is at (-2, 0).
The general equation of a circle with center (h, k) and radius r is:
(x - h)² + (y - k)² = r²
In this case, the center is (-2, 0), so we have:
(x + 2)² + (y - 0)² = r²
Now, we need to find the radius (r). Since the center is tangent to the x-axis, the distance from the center (-2, 0) to any point on the x-axis is the radius.
The distance from (-2, 0) to (0, 0) (a point on the x-axis) is 2 units. So, the radius (r) is 2.
Substitute the radius (r = 2) into the equation:
(x + 2)² + (y - 0)² = 2²
Simplify:
(x + 2)² + y² = 4
To express it in standard form, expand (x + 2)²:
(x² + 4x + 4) + y² = 4
Rearrange the terms:
x² + y² + 4x + 4 - 4 = 0
Simplify further:
x² + y² + 4x = 0
So, the equation of the circle in standard form is (x + 1)² + (y - 2)² = 5.