Determining the Equation of a Circle Given Five Points

It is a common misconception that three points are required to define a unique circle. While it is true that three non-collinear points define a unique circle, what happens when five points are provided? Can a circle be determined if it passes through all five points?

When a circle passes through five points, the approach is to first select three points and use them to determine the circle's equation. If the other two points do not lie on this circle, then the five points cannot be on a single circle. This is because three non-collinear points define a unique circle, and adding extra points without them lying on this circle would violate the definition of a circle.

Using Perpendicular Bisectors to Determine a Circle

To determine a circle from three points, you can use the method of drawing perpendicular bisectors of the chords formed by pairs of these points. The point at which these bisectors intersect is the center of the circle. Once the center is known, the radius can be found by measuring the distance from the center to any of the three points.

Given three points ( P_1(x_1, y_1) ), ( P_2(x_2, y_2) ), and ( P_3(x_3, y_3) ), the equations of the perpendicular bisectors can be derived, and their intersection will give the coordinates ((h, k)) of the circle's center. The radius ( r ) can be calculated as:

[ r sqrt{(h - x_1)^2 (k - y_1)^2} ]

With the center ((h, k)) and the radius ( r ), the equation of the circle can be written as:

[ (x - h)^2 (y - k)^2 r^2 ]

If the other two points ( P_4(x_4, y_4) ) and ( P_5(x_5, y_5) ) also satisfy this equation, then the five points are indeed on the same circle. Otherwise, they do not form a unique circle.

Equation of a Circle in the Cartesian Plane

In the Cartesian plane, a circle can be described by the equation:

[ (x - h)^2 (y - k)^2 r^2 ]

where ((h, k)) is the center of the circle and ( r ) is the radius. If five points ((x_1, y_1)), ((x_2, y_2)), ((x_3, y_3)), ((x_4, y_4)), and ((x_5, y_5)) are given, you can substitute the coordinates of three points into the equation to find ( h ), ( k ), and ( r ).

From the three equations:

[(x_1 - h)^2 (y_1 - k)^2 r^2] [(x_2 - h)^2 (y_2 - k)^2 r^2] [(x_3 - h)^2 (y_3 - k)^2 r^2]

solving these equations will give the values of ( h ), ( k ), and ( r ).

Checking Whether the Other Points Lie on the Circle

Once the center and radius are determined, you can substitute the coordinates of the other two points into the circle's equation to check if they lie on the circle. If both points satisfy the equation, then the five points form a unique circle. If not, the points do not lie on a single circle.

For example, checking the points ( P_4 ) and ( P_5 ) against the circle's equation:

[(x_4 - h)^2 (y_4 - k)^2 r^2] [(x_5 - h)^2 (y_5 - k)^2 r^2]

If both equations hold true, then the fifth points ( P_4 ) and ( P_5 ) are also on the circle.

By following these steps, you can determine whether a unique circle can be defined by five given points or not.

Conclusion

While three points are sufficient to uniquely define a circle, adding more points requires verification to ensure they all lie on the same circle. Utilizing the methods of perpendicular bisectors or direct substitution into the circle's equation provides a clear and systematic approach to this task. Understanding these techniques enhances your ability to work with geometric shapes and their equations in various applications, from theoretical mathematics to practical engineering.