SOLUTION: Please help me find the general form of the equation of the circle With endpoints of a diameter at (-6,-2) and (-4,4)

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Question 472993: Please help me find the general form of the equation of the circle
With endpoints of a diameter at (-6,-2) and (-4,4)
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 (-6,-2) and (-4,4), we need to find the midpoint.


X-Coordinate of Midpoint = %28x%5B1%5D%2Bx%5B2%5D%29%2F2+=+%28-6%2B-4%29%2F2=-10%2F2+=+-5


Since the x coordinate of midpoint is -5, this means that h=-5


Y-Coordinate of Midpoint = %28y%5B1%5D%2By%5B2%5D%29%2F2+=+%28-2%2B4%29%2F2=2%2F2+=+1


Since the y coordinate of midpoint is 1, this means that k=1


So the center is the point (-5, 1)


---------------------------------------------------


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=-5 and k=1. Also, x=-6 and y=-2 (drawn from one of the given points -- namely the first one). Plug these values into the equation above and simplify to get:


r%5E2=%28-6--5%29%5E2%2B%28-2-1%29%5E2


r%5E2=%28-6%2B5%29%5E2%2B%28-2-1%29%5E2


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


r%5E2=1%2B9


r%5E2=10


So because h=-5, k=1, and r%5E2=10, this means that the equation of the circle that passes through the points (-6,-2) and (-4,4) (which are the endpoints of the diameter) is


%28x%2B5%29%5E2%2B%28y-1%29%5E2=10.


So the answer is %28x%2B5%29%5E2%2B%28y-1%29%5E2=10