Determining Points on a Circle with Center (1, 1) and Radius 5

When dealing with circles in coordinate geometry, it's often necessary to identify points that lie on the circle's circumference. Let's explore the method to determine if a given point lies on a circle with a specific center and radius.

Understanding the Circle's Equation

The equation of a circle is given by (x-h)^2 (y-k)^2 r^2, where (h, k) is the center of the circle and ( r ) is the radius of the circle.

Applying the Equation to Our Circle

In this problem, the center of the circle is (1, 1) and the radius is 5. Plugging these values into the general circle equation, we get:

(x-1)^2 (y-1)^2 5^2

This simplifies to:

(x-1)^2 (y-1)^2 25

Checking Points to See if They Lie on the Circle

To determine if a point (x, y) lies on the circle, you substitute the coordinates of the point into the equation and check if the equation holds true. If the left-hand side equals 25, the point is on the circle; otherwise, it is not.

Examples of Points on the Circle

Let's test some points by substituting them into the equation:

For (1, 6): (1-1)^2 (6-1)^2 0 25 25 Result: Point (1, 6) is on the circle. For (1, -4): (1-1)^2 (-4-1)^2 0 25 25 Result: Point (1, -4) is on the circle. For (6, 1): (6-1)^2 (1-1)^2 25 0 25 Result: Point (6, 1) is on the circle. For (-4, 1): (-4-1)^2 (1-1)^2 25 0 25 Result: Point (-4, 1) is on the circle.

Alternative Methods to Verify the Points

Another method is to calculate the distance from the center of the circle to the point, which should be equal to the radius if the point is on the circle. Here's how you can do it:

The distance formula is given by:

d sqrt{(x_2 - x_1)^2 (y_2 - y_1)^2}

Using this formula with the center (1, 1) and the point (x, y), the distance should be 5 if the point is on the circle.

Example Calculation Using Distance Method

For the point (1, 6):

d sqrt{(1-1)^2 (6-1)^2} sqrt{0 25} sqrt{25} 5

This confirms that the point (1, 6) is on the circle.

Repeat this process for each point to verify.

Conclusion

To summarize, determining if a point lies on a circle involves checking whether it satisfies the circle's equation or verifying the distance from the center to the point. Mastering these methods enhances your understanding and problem-solving skills in coordinate geometry.