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QUESTION: i have a problem with figuring out the equation of a circle that (3,0) - (6,9) - (9,-3) are on it and they fulfill the the circle equation by using the general form of circle equation :
x^2+y^2+ax+by+c=0
when i tried solving it i got
3a+c=-9
6a+9b+c=-117
9a-3b+c=-90
but i tried a lot to solve the equations all the answers i got didn't give me 0 when i put them in the equation i got a=-18 b=-6 c =45
can you check it for me?
ANSWER: Note that 3a + c = 3(-18) + 45 = -54 + 45 = -9. 1st one checks.
)
Also, 6(-18) + 9(-6) + 45 = -108 - 54 + 45 = -117. 2nd one checks.
However, 9(-18) - 3(-6) + 45 = -162 + 18 + 45 = -162 + 63 = -99.
Oops.
See attached sheet for correct answer.
---------- FOLLOW-UP ----------
QUESTION: umm i did not understand how you solved it this isnt algebra right? I mean i was only taught to solve equations with algebra i don't understand all these tables
OK, I will do it over, since I don't think what I did was even right.
I mean, it had the right approach, but I have came up with the wrong answer.
The equations area
[A] 3a + c = -9,
[B] 6a + 9b + c = -117, and
[C] 9a - 3b + c = -90.
Now to solve a set of equations and unknowns, make the coefficient of the variable in that equation equal to 1 and add the appropriate multiple of that equation to the others in order to eliminate the coefficient of that variable. Please remember this and continue on the understand.
The [A], [B], and [C] are the equations.
The variables are a, b, and c.
To solve, get a 1 on the a in the first equation.
Cancel a from the equations below by doing the following:
The new [B] will be [B]-2[A].
The new [C] will be [C]-3[A].
This gives us
[A] a + c/3 = -3,
[B] 9b - c = -99, and
[C] -3b - 2c = -63.
Leave [A] alone this time.
Divide equation [B] by 9.
Compute the new [C] = [C] + [B]/3.
This gives us
[A] a + c/3 = -3,
[B] b - c/9 = -11, and
[C] -7c/3 = -96.
Now for the last operation.
Replace [A] with [A] + [C]/7.
Replace [B] with [B] - [C]/21.
Replace [C] with -3[C]/7.
That gives us
[A] a = -117/7,
[B] b = -45/7, and
[C] c = 288/7.
If we check these, hopefully they are right this time.
Our original equations to put the variables back in were
[A] 3a + c = -9,
[B] 6a + 9b + c = -117, and
[C] 9a - 3b + c = -90.
This gives us
[A] 3(-117/7) + 288/7 = -351/7 + 288/7 = -63/7 = -9. Check.
[B] 6(-117/7) + 9(-45/7) + 288/7 = (-702 - 405 + 288)/7 = -819/7
= -117. Check.
[C] 9(-117/7) - 3(-45/7) + 288/7 = (-1053 + 135 + 288)/7 = -630/7
= -90. Check.
Aha! The right answer came!
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