SOLUTION: A and B are the points of intersection of the circle with equation x^2 + y^2 + 4x - 6y + 9 =0 and the diameter passing through origin. Find the coordinates of A and B.

Algebra ->  Circles -> SOLUTION: A and B are the points of intersection of the circle with equation x^2 + y^2 + 4x - 6y + 9 =0 and the diameter passing through origin. Find the coordinates of A and B.       Log On


   

Question 768566: A and B are the points of intersection of the circle with equation x^2 + y^2 + 4x - 6y + 9 =0 and the diameter passing through origin. Find the coordinates of A and B.
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
A and B are the points of intersection of the circle with equation x^2 + y^2 + 4x - 6y + 9 =0 and the diameter passing through origin. Find the coordinates of A and B.
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Find the center and radius of the circle.
x^2 + y^2 + 4x - 6y + 9 =0
x^2 + y^2 + 4x - 6y = -9
%28x%2B2%29%5E2+%2B+%28y-3%29%5E2+=+4
Center at (-2,3), r = 2
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Equation of the line thru the center and the Origin is
y = -3x/2
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Sub for y in the eqn of the circle.
x^2 + y^2 + 4x - 6y + 9 = 0
x%5E2+%2B+%289x%5E2%29%2F4+%2B+4x+%2B+9x+%2B+9+=+0
13x%5E2%2F4+%2B+13x+%2B+9+=+0
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 3.25x%5E2%2B13x%2B9+=+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=%2813%29%5E2-4%2A3.25%2A9=52.

Discriminant d=52 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-13%2B-sqrt%28+52+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%2813%29%2Bsqrt%28+52+%29%29%2F2%5C3.25+=+-0.890599607549542
x%5B2%5D+=+%28-%2813%29-sqrt%28+52+%29%29%2F2%5C3.25+=+-3.10940039245046

Quadratic expression 3.25x%5E2%2B13x%2B9 can be factored:
3.25x%5E2%2B13x%2B9+=+%28x--0.890599607549542%29%2A%28x--3.10940039245046%29
Again, the answer is: -0.890599607549542, -3.10940039245046. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+3.25%2Ax%5E2%2B13%2Ax%2B9+%29

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x+=+-2+%2B-sqrt%28208%29%2F13
Use y = -3x/2 to find the y's of A and B