Given three points on a circle, find the equation in standard form.

Question

(1,1) (1,-5) (-3,3).
What do the three points have in common?
Are all three points the same distance from the center?
What does that different represent with respect to the circle?

Answer 1

The fact that the first two points have the same x-value indicates that they must be the endpoints of the diameter. We can find the center of the circle by finding the midpoint between those two points.

 

Midpoint: (1+1)/2, (-5+1)/2 --> (1, -2)

 

So the center of the circle is at (1, -2)

 

Next, we need to find the radius. To do this, we use the distance formula from the center to one of the points.

 

d= sqrt( (-2-1)^2 + (1-1)^2 ), which comes out to be 3. Thus, the radius is 3.

 

Standard equation of a circle is (x-h)^2 + (y-k)^2 = r^2 where (h,k) is the center and r is the radius.

 

Put it all together in the standard form equation of a circle with: (x-1)^2 + (y+2)^2 = 9