Find the equation of the circle passing through:
(3,-2) (4, 5) and (-6, -3)
To find the equation of the circle passing through the given points, we can first find the center of the circle by finding the intersection point of the perpendicular bisectors of any two of the line segments formed by the three points.
Let's first find the midpoints and slopes of the line segments:
1. Midpoint of (3,-2) and (4,5):
x1 = 3, y1 = -2
x2 = 4, y2 = 5
Midpoint:
x = (x1 + x2) / 2
x = (3 + 4) / 2
x = 3.5
y = (y1 + y2) / 2
y = (-2 + 5) / 2
y = 1.5
2. Midpoint of (4,5) and (-6,-3):
x1 = 4, y1 = 5
x2 = -6, y2 = -3
Midpoint:
x = (x1 + x2) / 2
x = (4 + (-6)) / 2
x = -1
y = (y1 + y2) / 2
y = (5 + (-3)) / 2
y = 1
Now, we can find the slope of the line segments:
1. Slope of line passing through (3,-2) and (4,5):
m = (5 - (-2)) / (4 - 3)
m = 7 / 1
m = 7
For the perpendicular bisector of the line segment passing through (3,-2) and (4,5), the slope will be -1/7.
2. Slope of line passing through (4,5) and (-6,-3):
m = (-3 - 5) / (-6 - 4)
m = -8 / -10
m = 4 / 5
For the perpendicular bisector of the line segment passing through (4,5) and (-6,-3), the slope will be -5/4.
Now, let's find the equations of the perpendicular bisectors:
1. For the line passing through (3.5, 1.5) with slope -1/7:
y - 1.5 = -1/7(x - 3.5)
y - 1.5 = -1/7x + 1/7(3.5)
y = -1/7x + 6/7 + 1.5
y = -1/7x + 6/7 + 10/7
y = -1/7x + 16/7
2. For the line passing through (-1, 1) with slope -5/4:
y - 1 = -5/4(x + 1)
y - 1 = -5/4x - 5/4
y = -5/4x - 5/4 + 1
y = -5/4x - 5/4 + 4/4
y = -5/4x - 1/4
Now, let's find the point of intersection of these two lines, which will be the center of the circle:
-1/7x + 16/7 = -5/4x - 1/4
-1/7x + 5/4x = -1/4 - 16/7
(5/4 - 1/7)x = (-7 - 64)/28
(35 - 4)x = -71/28
31x = -71/28
x = -71 / (31 * 28)
x = -71 / 868
x = -71 / (-4 * 217)
x = 71 / 868
x = 71 / 868
y = -1/4x - 1/4
y = -1/4(71/868) - 1/4
y = -71/3472 - 1/4
y = -71/3472 - 871/3472
y = -942/3472
y = -471/1736
The center of the circle is C(71/868, -471/1736).
To find the radius, we use one of the given points. Let's use (3, -2):
r^2 = (3 - 71/868)^2 + (-2 + 471/1736)^2
r^2 = (260/868)^2 + (-347/1736)^2
r^2 = 67600 / 753664 + 120409 / 3026416
r^2 = (67600 * 3026416 + 753664 * 120409) / 2284469395456
r^2 = 208920576 + 91035193696 / 2284469395456
r^2 = 91224068872 / 2284469395456
The equation of the circle is:
(x - 71/868)^2 + (y + 471/1736)^2 = 91224068872 / 2284469395456