Question

Circle O, with center (x, y), passes through the points A(0, 0), B(–3, 0), and C(1, 2). Find the coordinates of the center of the circle.

Answer 1

Since we are given three points on the circle, and we know that each point is a distance equal to the radius to some point (x,y), we can set up a system of equations to solve for the center...

Knowing that the distance between any two points is:

d^2=(x2-x1)^2+(y2-y1)^2 and these distances are all equal we can say

d^2=(x-0)^2+(y-0)^2, (x+3)^2+(y-0)^2, (x-1)^2+(y-2)^2

d^2=x^2+y^2, x^2+6x+9+y^2, x^2-2x+1+y^2-4y+4 now getting differences

0=-6x-9, 8x+4+4y

Since both of the equations above are equal to zero we can see that:

-6x-9=0

-6x=9

x=-9/6

x=-1.5, making 8x+4+4y=0 become:

8(-1.5)+4+4y=0

-12+4+4y=0

-8+4y=0

4y=8

y=2

So the center of this circle is at the point (-1.5, 2)