Write the equations of the circle in general form.
1. Points on circle (0,0), (0,8), (6,0)
2. Points on circle (1, -1), (2, -2), (0,-2).
I don't know how to solve this given the fact that I don't know what the center or the radius of the circle is.
points with the same x or y coordinate represent vertical or horizontal chords of the circle
the perpendicular bisectors of the chords are diameters, and they give one of the coordinates of the center
you can use the distance formula to find the other coordinate of the center
1. the center of the circle is (x,4)
2. the center of the circle is (1,y)
To find the general form of a circle, you need the center coordinates and the radius. However, given the points on the circle, we can still find the equation of the circle using an alternate approach. Let's go through both cases:
1. Points on circle (0,0), (0,8), (6,0):
Step 1: Find the midpoint of any two given points. Let's consider the points (0,0) and (0,8) as the center of the circle. The midpoint formula is given by:
x-coordinate of midpoint = (x1 + x2) / 2
y-coordinate of midpoint = (y1 + y2) / 2
Substituting the values, we get:
x-coordinate of midpoint = (0 + 0) / 2 = 0
y-coordinate of midpoint = (0 + 8) /2 = 4
So, the center of the circle is (0, 4).
Step 2: Find the radius of the circle. The radius is the distance between the center and any given point on the circle. Let's consider the point (0,8).
The distance formula is given by:
r = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Substituting the values, we get:
r = sqrt((0 - 0)^2 + (8 - 4)^2) = sqrt(0 + 16) = 4
So, the radius of the circle is 4.
Step 3: Use the general form of the circle equation, which is (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center and r is the radius.
Substituting the values, we get:
(x - 0)^2 + (y - 4)^2 = 4^2
x^2 + (y - 4)^2 = 16
Therefore, the equation of the circle is x^2 + (y - 4)^2 = 16.
2. Points on circle (1, -1), (2, -2), (0,-2):
Following the same steps:
Step 1: Midpoint of (1, -1) and (2, -2):
x-coordinate of midpoint = (1 + 2) / 2 = 3/2
y-coordinate of midpoint = (-1 - 2) / 2 = -3/2
So, the center of the circle is (3/2, -3/2).
Step 2: Radius using the distance formula and the point (0,-2):
r = sqrt((2 - 3/2)^2 + (-2 + 3/2)^2) = sqrt(1/4 + 1/4) = sqrt(1/2)
So, the radius of the circle is sqrt(1/2).
Step 3: Using the general form, the equation of the circle is:
(x - 3/2)^2 + (y + 3/2)^2 = (sqrt(1/2))^2
(x - 3/2)^2 + (y + 3/2)^2 = 1/2
Therefore, the equation of the circle is (x - 3/2)^2 + (y + 3/2)^2 = 1/2.
Note that these equations are in general form for circles.