Write the standard form of the equation of a circle that passes through the points at (0,8) (8,0) and (16,8). Then identify the center and radius of the circle.
(x-8)^2 + (y-8)^2 = 64
(x^2 / 64) + (y^2 / 64) = 64
center=(8,8)
radius=8
I got it wrong don't know how
(x-8)^2 + (y-8)^2 = 64
fine
(x^2 / 64) + (y^2 / 64) = 64
but where did you get this??????
center=(8,8)
fine
radius=8
fine
You will have to write the equation
(x-h)^2+(y-k)^2=r^2
for the three sets of points given. Then solve for h, k, r
It is ugly but solvable.
To find the standard form of the equation of a circle that passes through three given points, we can use the following steps:
1. Calculate the perpendicular bisectors of two of the line segments formed by the three points.
2. The point where the two perpendicular bisectors intersect will be the center of the circle.
3. Calculate the distance between the center and any of the three given points to find the radius.
Let's go through these steps:
1. Calculate the perpendicular bisectors of two of the line segments formed by the three points:
a) The line segment between (0, 8) and (8, 0):
- The midpoint of the line segment is [(0 + 8) / 2, (8 + 0) / 2] = (4, 4).
- The slope of the line segment is (0 - 8) / (8 - 0) = -1.
- Therefore, the slope of the perpendicular bisector is the negative reciprocal of -1, which is 1.
- Using the point-slope form of a line, the equation of the perpendicular bisector is y - 4 = 1(x - 4), which simplifies to y = x.
b) The line segment between (0, 8) and (16, 8):
- The midpoint of the line segment is [(0 + 16) / 2, (8 + 8) / 2] = (8, 8).
- The slope of the line segment is (8 - 8) / (16 - 0) = 0.
- For a line with a slope of 0, the perpendicular bisector is a horizontal line passing through the midpoint.
- Using the slope-intercept form, the equation of the perpendicular bisector is y = 8.
2. Find the intersection point of the two perpendicular bisectors:
- Since one of the perpendicular bisectors is a horizontal line (y = 8) and the other is y = x,
we can substitute y = 8 into y = x to find the x-coordinate of the intersection point.
x = 8.
- Substituting x = 8 into y = x, we find that y = 8 as well.
- Therefore, the intersection point is (8, 8).
3. Calculate the distance between the center and any of the given points to find the radius:
- We can use the distance formula: distance = sqrt((x2 - x1)^2 + (y2 - y1)^2).
- Let's calculate the distance between the center (8, 8) and one of the given points, (0, 8):
radius = sqrt((0 - 8)^2 + (8 - 8)^2) = sqrt((-8)^2 + 0) = sqrt(64) = 8.
Therefore, the standard form of the equation of the circle passing through the points at (0, 8), (8, 0), and (16, 8) is:
(x - 8)^2 + (y - 8)^2 = 64.
The center of the circle is at (8, 8), and the radius is 8.