SOLUTION: Find the equation of the circle that passes through the points of intersection of the circles x²+y²=2x, x²+y²=2y and has its center on the line y=2.

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Question 1143657: Find the equation of the circle that passes through the points of intersection of the circles x²+y²=2x, x²+y²=2y and has its center on the line y=2.
Found 2 solutions by ikleyn, Alan3354:
Answer by ikleyn(52802)   (Show Source): You can put this solution on YOUR website!
.
(1)  Since the left sides are identical in given equations of the circles, their right sides are equal for intersection points

         x = y.



(2)  Substituting  x=y  into the circles equations gives

         x^2 + x^2 = 2x  ====>  2x^2 = 2x  ====>  2x^2 - 2x = 0  ====>  2x(x-1) = 0.


     The roots are  x= 1  and  x= 0,  so the intersection points are  A=(1,1)  and  B=(0,0).



(3)  The center of the third circle lies on the perpendicular bisector to the segment AB.


     An equation of this perpendicular bisector is  y-0.5 = -(x-0.5) = -x + 0.5.



(4)  At the same time, the center of the third circle lies on the line  y= 2.


     So, the center of the third circle is at

         2 - 0.5 = -x + 0.5,

     which implies  x= -1.


     Thus  the center of the third circle is the point  P = (-1,2).


     The radius of the third circle is the distance of the point P from the origin (0,0), i.e.   =  = .


(5)  Thus the equation of the circle under the question is


          +  = ,

     or, equivalently,


          +  = 5.    ANSWER


Answer by Alan3354(69443)   (Show Source): You can put this solution on YOUR website!
Find the equation of the circle that passes through the points of intersection of the circles x²+y²=2x, x²+y²=2y and has its center on the line y=2.
---------------------
Find the 2 points of intersection.
They're (0,0) and (1,1)
=============================
The center is on the perpendicular bisector of y=x thru the 2 points ---> slope = -1 thru the point (1/2,1/2)
---> y - 0.5 = -1*(x - 0.5)
y = -x + 1
==============
The center is the intersection of y = -x + 1 and y = 2
2 = -x + 1
x = -1
y = 2
---> the center is (-1,2)
--------------------
r = the distance from (-1,2) to either point.
r = sqrt(5)
The circle is




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