Math CentralQuandaries & Queries


Question from Cristela, a student:

find the equation and all the information in General Form and Standard Form of the Circle that will passed trough the point (2,3) (6,1) (4,-3)

Hi Cristela.

If you know the radius and the center of the circle, you can quickly plug that into the standard form for a circle and then generate the General form, right? I'll assume that isn't your problem, but that you are instead having trouble finding the radius and center.

One way to solve it:
Remember that with circle geometry, and two distinct points on the circumference create a chord when joined. The perpendicular bisector of a chord always passes through a circle's center. Therefore, construct a chord using two known points on a line. Then find the midpoint of the chord by simply averaging the x and y values of the two points. Then construct an equation of the perpendicular bisector using the midpoint and the slope (the slope of the perpendicular bisector is the negative inverse of the slope of the chord). Do this for two different chords. Then you'll have two lines whose intersection is the center of the circle.

Once you have the center of the circle, you can use the distance formula to find the radius and you can apply this to the standard form of the circle.

Another approach:
Use three simultaneous equations for each of the three points, all of which are in standard circular form and involve the same values of (a, b) and r (the center and the radius of the circle). Given three equations and three unknowns, you can use the elimination and substitution methods for solving simultaneous equations to quickly find (a,b) and r, then you can create a standard form equation of the resulting circle and go from there to the general form.

Hope this helps,
Stephen La Rocque

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