SOLUTION: Find the equation of the circle in the form X^2+y^2+Dx+Ey+F=0 of the circle through points (1,1) (2,-1), (2,3)

Question 771638: Find the equation of the circle in the form X^2+y^2+Dx+Ey+F=0 of the circle through points (1,1) (2,-1), (2,3)

Found 2 solutions by josgarithmetic, MathLover1:
Answer by josgarithmetic(39620) (Show Source): You can put this solution on YOUR website!
The actual variables will be D, E, and F. The x and y values will become constants or part of constants. The system of equations will be more comfortable with each equation in the form, .

Each point gives one equation:
SYSTEM:

'
SIMPLIFIED SYSTEM:
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You can solve for the three variables any way you know or like. You can also solve using the matrix,

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Intentionally unfinished allowing you to finish yourself.

Answer by MathLover1(20850) (Show Source): You can put this solution on YOUR website!
Consider the general equation of a circle as .
given: points (,) (,), (,)
Using the given three points we derive our equation :
From the point (,) we get the equation
or
........ equation 1
From the point (,) we get
or
........ equation 2

From the point (,) we get
or
....... equation 3

We now have a system of three equations :

........ equation 1
........ equation 2
....... equation 3

We now use elimination method to find the value of , and .
Using equation 1 and eq. 2 to eliminate , we do this by subtracting equation 2 from eq. 1 :

..... let this be equation 4

Using Equation 2 and eqn 3, if we subtract eq. 3 from eq. 2 we can eliminate and :

=>
go back to eq.4 and plug in
.......solve for

go back to eq.1 and plug in and and solve for
........ equation 1

We get this general equation by substituting
, and to the general equation of the circle
:

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