a circle passes through the points (6,-6) (3, -7) and (3, 3). Find the center and radius of the circle.
+129899 Call the center ( x, y)
We have these 3 equations
(x - 6)^2 + ( y + 6)^2 = r^2 (1)
(x - 3)^2 + ( y + 7)^2 = r^2 (2)
(x - 3)^2 + (y - 3)^2 = r^2 (3)
Subtract the third equation from the second and we have
(y + 7)^2 - (y - 3)^2 = 0 simplify
y^2 + 14y + 49 - y^2 + 6y - 9 = 0
20y + 40 = 0
20y = - 40
y = -40 / 20 = - 2
Replace this value for y in the first two equations and simplify
( x-6)^2 + ( -2 + 6)^2 = r^2 ⇒ x^2 - 12x + 36 + 16 = r^2 ⇒ x^2 -12x + 52 = r^2 ( 4)
(x - 3)^2 + (-2 + 7)^2 = r^2 ⇒ x^2 - 6x + 9 + 25 = r^2 ⇒ x^2 - 6x + 34 = r^2 (5)
Subtract (5) from (4)
-6x + 18 = 0
-6x = -18
x = -18 / -6 = 3
The center is (3, -2) and to find the radius we can use (1)
(3-6)^2 + (-2 + 6)^2 = r^2
9 + 16 = r^2
25 = r^2 take the positive sqrt
5 = r
Here's a graph : https://www.desmos.com/calculator/hwro781e8y
