Final answer:
- The coordinates of the center of the circle are (1, 7).
- The length of the radius is √10.
- The equation of the circle in standard form is (x - 1)^2 + (y - 7)^2 = 10
- The equation of the circle in general form is x^2 + y^2 - 2x - 14y + 40 = 0.
Step-by-step explanation:
To find the center of the circle, we can use the midpoint formula. The midpoint formula states that the coordinates of the midpoint of a line segment with endpoints (x1, y1) and (x2, y2) are given by:
(x, y) = ((x1 + x2)/2, (y1 + y2)/2)
In this case, the coordinates of the endpoints of the diameter are (4, 8) and (-2, 6). We can plug these values into the formula to find the center:
(x, y) = ((4 + -2)/2, (8 + 6)/2)
(x, y) = (1, 7)
So, the coordinates of the center of the circle are (1, 7).
To determine the length of the radius, we can use the distance formula. The distance formula states that the distance between two points (x1, y1) and (x2, y2) is given by:
d = √((x2 - x1)^2 + (y2 - y1)^2)
In this case, we can use the distance formula to find the distance between the center of the circle (1, 7) and one of the endpoints of the diameter (4, 8):
d = √((4 - 1)^2 + (8 - 7)^2)
d = √(9 + 1)
d = √10
So, the length of the radius is √10.
To write the equation of the circle in standard form, we can use the formula:
(x - h)^2 + (y - k)^2 = r^2
In this case, the coordinates of the center are (1, 7) and the length of the radius is √10. Plugging these values into the formula, we get:
(x - 1)^2 + (y - 7)^2 = 10
So, the equation of the circle in standard form is (x - 1)^2 + (y - 7)^2 = 10.
To write the equation of the circle in general form, we can expand and simplify the equation:
(x - 1)^2 + (y - 7)^2 = 10
x^2 - 2x + 1 + y^2 - 14y + 49 = 10
x^2 + y^2 - 2x - 14y + 40 = 0
So, the equation of the circle in general form is x^2 + y^2 - 2x - 14y + 40 = 0.