Determining if a Point Lies on the Boundary of a Circle

When working with geometric problems, it's crucial to understand the relationship between points and circles. Specifically, determining if a point lies on the boundary (circumference) of a circle involves using the circle's equation and the distance formula. Let's explore a detailed explanation and example to illustrate this concept.

Example: Is the Point (3, 3√3) on the Boundary of the Circle x2 y2 36?

The equation x2 y2 36 represents a circle centered at the origin (0, 0) with a radius of 6 units. To determine if the point (3, 3√3) lies on the boundary of this circle, we need to substitute the coordinates of the point into the circle's equation and evaluate the result.

Step-by-Step Solution:

Substitute the coordinates (3, 3√3) into the equation x2 y2 36.

32   (3√3)2  9   27  36

The left-hand side simplifies to 36, which is equal to the right-hand side of the equation (36).

Interpret the result. Since 32 (3√3)2 36, the point (3, 3√3) satisfies the equation of the circle. This indicates that the point lies on the boundary (circumference) of the circle.

General Method for Checking Point Location in a Circle

In general, to determine the location of a point (x1, y1)

in relation to a circle with the equation x2 y2 r2", you can follow these steps:

Substitute the coordinates (x1, y1) into the circle's equation.

Evaluate the expression x12 y12.

Compare the result with r2, the radius squared of the circle.

If x12 y12 r2, the point is inside the circle. If x12 y12 r2, the point is on the boundary (circumference) of the circle. If x12 y12 r2, the point is outside the circle.

Additional Example

Consider the circle with the equation x2 y2 9. We want to check if the point (3, 3) lies on the boundary of this circle.

Substitute the coordinates (3, 3) into the equation:

32   32  9   9  18

Compare the result with 9:

Since 18 9, the point (3, 3) is outside the circle.

By applying the same method, we can determine the location of any point with respect to a given circle.

Conclusion

Understanding the relationship between points and circles is essential in many areas of mathematics and its applications. By following the steps outlined in this article, you can quickly and easily determine if a point lies on the boundary of a circle or not. If you have questions or further examples, feel free to explore or seek additional resources.