SOLUTION: A circle passes through (3,5) and(1,2) ,find its equation

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Question 551091: A circle passes through (3,5) and(1,2) ,find its equation
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
I'm assuming that these are the endpoints of the diameter.


Recall that the general equation of a circle is %28x-h%29%5E2%2B%28y-k%29%5E2=r%5E2.


So we need the center (h,k) and the radius squared r%5E2.


First, let's find the center (h,k).


Since the center is the midpoint of the line segment with endpoints (3,5) and (1,2), we need to find the midpoint.


X-Coordinate of Midpoint = %28x%5B1%5D%2Bx%5B2%5D%29%2F2+=+%283%2B1%29%2F2=4%2F2+=+2


Since the x coordinate of midpoint is 2, this means that h=2


Y-Coordinate of Midpoint = %28y%5B1%5D%2By%5B2%5D%29%2F2+=+%285%2B2%29%2F2=7%2F2


Since the y coordinate of midpoint is 7%2F2, this means that k=7%2F2


So the center is the point (2, 7/2)


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Now let's find the radius squared


Use the formula r%5E2=%28x-h%29%5E2%2B%28y-k%29%5E2, where (h,k) is the center and (x,y) is an arbitrary point on the circle.


In this case, h=2 and k=7%2F2. Also, x=3 and y=5. Plug these values into the equation above and simplify to get:


r%5E2=%283-2%29%5E2%2B%285-7%2F2%29%5E2


r%5E2=%281%29%5E2%2B%283%2F2%29%5E2


r%5E2=1%2B9%2F4


r%5E2=13%2F4


So because h=2, k=7%2F2, and r%5E2=13%2F4, this means that the equation of the circle that passes through the points (3,5) and (1,2) (which are the endpoints of the diameter) is


%28x-2%29%5E2%2B%28y-7%2F2%29%5E2=13%2F4.