๐ Finding Circle Equations: 2 Points vs. 3 Points
Let's explore the difference between finding the equation of a circle when you know two points on the circle versus when you know three points. Each scenario requires a different approach.
๐ฏ Definition of Finding Circle Equation from 2 Points
When you have only two points on the circle, you can't uniquely define a circle. Instead, the two points define a diameter. The center of the circle must lie on the perpendicular bisector of the line segment connecting the two points. You'll also need the radius which is half the length of the segment.
โญ๏ธ Definition of Finding Circle Equation from 3 Points
Given three non-collinear points, there exists a unique circle passing through all three points. These points define a system of equations that can be solved to find the center and radius of the circle. This often involves solving simultaneous equations.
๐ Comparison Table
| Feature |
2 Points |
3 Points |
| Uniqueness |
Infinite circles possible (diameter defined). |
Unique circle. |
| Method |
Find midpoint (center) and half the distance (radius). |
Solve a system of equations. |
| Equation Needed |
Midpoint formula, Distance formula |
General circle equation: $(x-h)^2 + (y-k)^2 = r^2$ |
| Complexity |
Relatively simpler. |
More complex, involving algebra. |
| Collinearity |
Not applicable. |
Points must be non-collinear. |
๐ Key Takeaways
- ๐ Two Points: With only two points, you can find many circles. These points define a diameter, and any circle with that diameter and centered on the perpendicular bisector will pass through those points.
- ๐งฎ Three Points: Three non-collinear points uniquely define a circle. The center and radius can be found by solving a system of equations derived from the general circle equation.
- ๐ก Simplification: Finding a circle equation from two points is generally simpler than from three points.
- ๐งญ Uniqueness Matters: The key difference lies in the uniqueness of the solution; two points offer infinite possibilities, while three points provide a single, specific circle.
- โ๏ธ System of Equations: With three points, you'll need to set up and solve a system of equations to pinpoint the circle's center and radius.
- ๐ Perpendicular Bisector: For two points, the center of the circle must lie on the perpendicular bisector of the line segment joining those points.
- โ Radius Calculation: When you have two points defining a diameter, the radius is simply half the distance between the two points.