SOLUTION: if a cirlce has (-2,-3) and (6,1) as endpoints of its diameter where does the circle cross the y-axis?

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Question 435599: if a cirlce has (-2,-3) and (6,1) as endpoints of its diameter where does the circle cross the y-axis?
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
Find the eqn of the circle:
The midpoint is the center, point C
Cx = (-2+6)/2 = 2
Cy = (-3+1)/2 = -1
Center @ (2,-1)
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The radius is the distance from C to either point
r%5E2+=+4%5E2+%2B+2%5E2+=+20
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%28x-2%29%5E2+%2B+%28y%2B1%29%5E2+=+20 is the circle
It crosses the x-axis when y = 0
x%5E2+-+4x+%2B+4+%2B+1+=+20
x%5E2+-+4x+-+15+=+0
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B-4x%2B-15+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-4%29%5E2-4%2A1%2A-15=76.

Discriminant d=76 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--4%2B-sqrt%28+76+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-4%29%2Bsqrt%28+76+%29%29%2F2%5C1+=+6.35889894354067
x%5B2%5D+=+%28-%28-4%29-sqrt%28+76+%29%29%2F2%5C1+=+-2.35889894354067

Quadratic expression 1x%5E2%2B-4x%2B-15 can be factored:
1x%5E2%2B-4x%2B-15+=+%28x-6.35889894354067%29%2A%28x--2.35889894354067%29
Again, the answer is: 6.35889894354067, -2.35889894354067. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-4%2Ax%2B-15+%29

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x = 2 +/- sqrt(19)