Discussion





This is a variation on the standard "given three non-collinear points,

determine the circle that intersects the the three points" problem.





The process involves several steps based on the geometric fact that the

perpendicular bisectors of all possible chords of a circle intersect in the

center of the circle.  A chord of a circle is a line segment that intersects

the circle in two places.





You need to determine the equations of the lines containing two chords, find

the mid-points of each chord, determine the equations of the perpendicular

bisectors of the two chords, and solve the system of equations represented by

the equations of the perpendicular bisectors.  That ordered pair solution will

be the circle center.





Equation of a line given two points:  





Mid-point formula:  (,)





Equation of a line given the slope and one point:  





For part b of the problem, the distance formula applied to the circle center

and any one of the given points will give the radius.





The distance formula: 







Solution





First chord:  (4,0) to (0,2)







Slope is 



x-coordinate of mid-point:  

y-coordinate of mid-point:  





First chord mid-point is (2,1)





Equation of perpendicular bisector.  Slope of perpendicular:  .  Needs to contain point (2,1)







 is the equation of the first perpendicular bisector.





Second chord:  (-4,0) to (0,2)







Slope is 



x-coordinate of mid-point:  

y-coordinate of mid-point:  





Second chord mid-point is (-2,1)





Equation of perpendicular bisector.  Slope of perpendicular:  .  Needs to contain point (-2,1)







 is the equation of the second perpendicular bisector.





System of equations:

 1st perpendicular bisector

 2nd perpendicular bisector





Add the equations:



















So the circle center is at (0,-3)





Problem part b:

Calculate the distance from (0,2) to (0,-3).  Since both of these points lie

on the y-axis, you can just subtract the two y-coordinates to determine the

distance representing the radius of the circle:  .  But in

general, you would use the distance formula:











 

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