SOLUTION: The two points (-2,4) and (4,2) are the endpoints of the diameter of a circle. What is the equation of this circle in standard form?

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Question 374780: The two points (-2,4) and (4,2) are the endpoints of the diameter of a circle. What is the equation of this circle in standard form?
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
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 (-2,4) and (4,2), we need to find the midpoint.


X-Coordinate of Midpoint = %28x%5B1%5D%2Bx%5B2%5D%29%2F2+=+%28-2%2B4%29%2F2=2%2F2+=+1


Since the x coordinate of midpoint is 1, this means that h=1


Y-Coordinate of Midpoint = %28y%5B1%5D%2By%5B2%5D%29%2F2+=+%284%2B2%29%2F2=6%2F2+=+3


Since the y coordinate of midpoint is 3, this means that k=3


So the center is the point (1, 3)


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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=1 and k=3. Also, x=-2 and y=4. Plug these values into the equation above and simplify to get:


r%5E2=%28-2-1%29%5E2%2B%284-3%29%5E2


r%5E2=%28-3%29%5E2%2B%281%29%5E2


r%5E2=9%2B1


r%5E2=10


So because h=1, k=3, and r%5E2=10, this means that the equation of the circle that passes through the points (-2,4) and (4,2) (which are the endpoints of the diameter) is


%28x-1%29%5E2%2B%28y-3%29%5E2=10.


If you need more help, email me at jim_thompson5910@hotmail.com

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Jim