Question 1076714: Find an equation of the circle that passes through the points (2,3), (4,5), and (0,-3).
Found 2 solutions by MathLover1, Alan3354:
Answer by MathLover1(20850)   (Show Source):
You can put this solution on YOUR website! Find an equation of the circle that passes through the points ( , ), ( , ), and ( , ).
The standard form equation of a circle is  where
 and  are the  and  coordinates of the center of the circle
first find coordinates of the center (,) using given points
The first step is to set up these 3 equations by plugging the x- and y-coordinates of the points
(,), (,), and (,) into the circle formula:
 .................eq1
 ..................eq2
 .................eq3
----------------------------------------------
 .................eq1
 ..................eq2
 ...............eq3
--------------------------------------------
.................eq1
..................eq2...............subtract eq2 from eq1
---------------------------------------------------------
 ............simplify
 .............eq1a
h^2 - 4h + k^2 - 6 k + 13 = r^2.................eq1
h^2 + k^2 + 6k + 9 = r^2...............eq3...........subtract eq3 from eq1
------------------------------------------
 ............divide by -4
 ..............................eq2a
use
 ...............eq1a
 ..............................eq2a
-------------------------------------------------------------subtract eq1a from eq2a
now find
...............eq1a
so, center is at ( ,)
use one point, h and k to find r
 .................eq1
and, your equation is:
(2,3), (4,5), and (0,-3).
Answer by Alan3354(69443)  (Show Source):
You can put this solution on YOUR website! Find an equation of the circle that passes through the points (2,3), (4,5), and (0,-3).
-----------
Use determinants.
For this circle:
|(x^2+y^2) x y 1|
| 13 2 3 1|
| 41 4 5 1| = 0
| 9 0 -3 1|
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You can make an Excel sheet to solve it.
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Or, a geometrical approach:
Find an equation of the circle that passes through the points A(2,3), B(4,5), and C(0,-3).
I labeled the points.
Find the perpendicular bisectors of AB and BC.
The intersection of the bisectors is the center of the circle, (h,k).
The distance from the center to any of the 3 points is the radius, r.
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