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Write the equation of the circle that has a diameter whose
endpoints are (5, 27) and (22, 4).
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3 Answers

I am new to this answers forum, and there seem to be limited tools for which I can use to write, but I'll try my best.
First you need the Standard Form of the Equation of a Circle:
(x-h)+ (y-k)2 = r2 
First you want to find the distance between the two endpoints, which will give you your Diameter, using the Distance Formula. Then divide that answer by two to get your Radius which you will plug directly into your Equation of a Circle.
The (h,k) is the Center Point of your circle. You can find this with with Midpoint Formula using the two endpoints given in the problem.
I solved this problem using the steps above and I got the answer:
(x-13.5)2+(y-15.5)2 = (14.3)2
Normally you would leave the h and k as fractions, but I was having trouble figuring out how to format it properly. 
Also if there is anything I missed, you disagree with, or something I did not explain fully, please tell me. 
d = √((27-4)^2 + (22- 5) ^2) =28.6
r = 28.6/2 = 14.3
Center ( (22 +5)/2 , (27+4)/2 ) = ( 27/2, 31/2)
 Equation of a circle  ( X - 27/2 ) ^2 + ( Y - 31/2) ^2 = (14.3 ) ^2= 204.50
To solve this problem, we will need to do some work behind the scenes, so we can end up with a result that looks like this:

(x-a)2 + (y-b)2 = r2

The first step in this process is understanding that we are given the endpoints of a line, which is the diameter of the circle we are describing.  All we need to do is to determine two things: the midpoint (a,b) and the radius.  Let's start with the midpoint:

Using the midpoint formula,

    ((½)(x1 + x2) , (½)(y1 + y2))

We can use our two endpoints to determine the midpoint (a,b), and the center of our circle.

  1. ((½)(5+ 22) , (½)(27+ 4))
  2. ((½)(27) , (½)(31))
  3. (13.5, 15.5)

Now that we have our midpoint, we can figure out the radius by determining the distance between our midpoint and any point on the edge of our circle.  Good thing we know at least two points!

Using the Standard Circle Equation,  

    (x-a)2 + (y-b)2 = r2

We can determine the radius r using either of the endpoints.

  1. (5-13.5)2 + (27- 15.5)2 = r2
  2. (-8.5)2 + (11.5)2 = r2
  3. 204.5 = r2

Now that we have our midpoint coordinate and our radius, just place them into the circle equation:

(x-13.5)2 + (y-15.5)2 = 204.5