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Circle Geometry - Equation of a circle through 3 points

Hey, I need help on this question:
Find the equation of the circle passing through the points (6,3), (-5,2), (7,2)

Thanks
The perpendicular bisector of a chord passes through the centre.
Original post by constellarknight
The perpendicular bisector of a chord passes through the centre.


I found the equation of lines AB/AC - by finding their midpoints/gradients and plugging them into the y-y1=m(x-x1) formula. I then intersected them to find coordinates x,y of the centre but it still wasn't the correct answer.
You want the perpendicular bisectors of AB and AC, not the lines AB and AC themselves. The intersection point of the bisectors should then give the centre if you do it correctly.
Original post by constellarknight
You want the perpendicular bisectors of AB and AC, not the lines AB and AC themselves. The intersection point of the bisectors should then give the centre if you do it correctly.


I did this but it's still incorrect.
Then the given answer is incorrect, or you've made an arithmetic/algebraic error. Post your working.
A(6,3), B(-5,2), C(7,2):

Midpoint of AB; (6-5/2, 3+2/2) = (1/2, 5/2)
m. of AB = 2-3/-5-6 = 1/11 ==> m2=-11
y-5/2=-11(x-1/2)
y-5/2=-11x+11/2
2y-5=-22x+11
2y=-22x+16
y=-11x+8 - (1)

Midpoint of AC; (7+6/2, 2+3/2) = (13/2, 5/2)
m.of AC= 2-3/7-6 = -1 ==> m2=1
y-5/2=(x-13/2)
y-5/2=x-13/2
2y-5=2x-13
2y=2x-8
y=x-4 - (2)

(1)=(2)
-11x+8 = x-4
8=12x-4
12x = 12 ==>x=1

Sub x=1 into 2;
y=1-4=-3

Okay, you were right about having an arithmetic error during this stage. However, I'm having problems on figuring out the radius. Advice would be really appreciated.
The radius is the distance from one of the points A, B, or C to the centre that you've just found.
Original post by constellarknight
The radius is the distance from one of the points A, B, or C to the centre that you've just found.


So, in this case it would be 122?
(edited 8 years ago)
(6,3) to (1,-3) = sqrt(25+36)=sqrt(61)
Hey, sorry but could you help me with this question - it is of the same topic;
Find the equations of the circle passing through these points:
(2,0), (3,1), (-6,10).
Oh hey, sorry I just realised that I have done another arithmetic error. Sorry! Thanks for your help though, your method has helped a lot! :smile:
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