SOLUTION: find the center,radius and standard form in general and standard form if the given circle passes through (0,6) (4,-2) (9,3) (Use general form)

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Question 1086015: find the center,radius and standard form in general and standard form if the given circle passes through (0,6) (4,-2) (9,3) (Use general form)
Found 2 solutions by Fombitz, Edwin McCravy:
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
Use this general form of a circle,
x%5E2%2Bax%2By%5E2%2Bby%2Bc=0
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(0,6)
0%5E2%2Ba%280%29%2B6%5E2%2Bb%286%29%2Bc=0
36%2B6b%2Bc=0
1.6b%2Bc=-36
(4,-2)
4%5E2%2Ba%284%29%2B%28-2%29%5E2%2Bb%28-2%29%2Bc=0
16%2B4a%2B4-2b%2Bc=0
2.4a-2b%2Bc=-20
(9,3)
9%5E2%2Ba%289%29%2B3%5E2%2Bb%283%29%2Bc=0
91%2B9a%2B9%2B3b%2Bc=0
3.9a%2B3b%2Bc=-90
Subtract 1 from 2,
4a-2b%2Bc-6b-c=-20%2B36
4a-8b=16
4.a-2b=4
Subtract 1 from 3,
9a%2B3b%2Bc-6b-c=-90%2B36
9a-3b=-54
5.3a-b=-18
Multiply 5 by -2 and add to 4,
a-2b-6a%2B2b=4%2B36
-5a=40
a=-8
Then,
-8-2b=4
-2b=12
b=-6
and
finally,
6%28-6%29%2Bc=-36
-36%2Bc=-36
c=0
So,
x%5E2%2By%5E2-8x-6y=0
%28x%5E2-8x%2B16%29%2B%28y%5E2-6y%2B9%29=16%2B9
highlight%28%28x-4%29%5E2%2B%28y-3%29%5E2=25%29
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.
.
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Answer by Edwin McCravy(20064) About Me  (Show Source):
You can put this solution on YOUR website!
find the center,radius and standard form in general and standard form if the given circle passes through (0,6) (4,-2) (9,3) (Use general form)
General for any circle:

x%5E2%2By%5E2%2BDx%2BEy%2BF=0

We substitute each point and simplify:

Substituting (x,y) = (0,6)

0%5E2%2B6%5E2%2BD%280%29%2BE%286%29%2BF=0

0%2B36%2B0%2BE%286%29%2BF=0

36%2B6E%2BF=0

6E%2BF=-36


Substituting (x,y) = (4,-2)

4%5E2%2B%28-2%29%5E2%2BD%284%29%2BE%28-2%29%29%2BF=0

16%2B4%2B4D-2E%2BF=0

20%2B4D-2E%2BF=0

4D-2E%2BF=-20

Substituting (x,y) = (9,3)

9%5E2%2B3%5E2%2BD%289%29%2BE%283%29%29%2BF=0

81%2B9%2B9D%2B3E%2BF=0

90%2B9D%2B3E%2BF=0

9D%2B3E%2BF=-90

Solve the system of equations:




Solve that system by elimination, substitution, 
matrices, or a combination of those.

Get D=-8, E=-6, F=0

x%5E2%2By%5E2-8x-6y%2B0=0

x%5E2%2By%5E2-8x-6y=0

Place in standard form

Switch the two middle terms:

x%5E2-8x%2By%5E2-6y=0

Multiply the coefficient of x by 1/2, get (1/2)(-8) = -4
Square -4, get (-4)˛ = +16
Add +16 after the second term, and add it to the right side.
Multiply the coefficient of y by 1/2, get (1/2)(-6) = -3
Square -3, get (-3)˛ = +9
Add +9 after the last term on the left, and add to right side.

x%5E2-8x%2B16%2By%5E2-6y%2B9=0%2B16%2B9

Factor first three terms:  x˛-8x+16 = (x-4)(x-4) = (x-4)˛
Factor last three terms on left:  y˛-6x+9 = (y-3)(y-3) = (y-3)˛
Combine numbers on the right sides; 0+16+9 = 25

%28x-4%29%5E2%2B%28y-3%29%5E2=25

Compare to

%28x-h%29%5E2%2B%28y-k%29%5E2=r%5E2

Center = (h,k) = (4,3) , radius˛ = r˛ = 25, r=5



Edwin