(x-1)^2+(y-8)^2=4

Jasonbradley

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Jun 17, 2024

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Understanding the Equation: (x-1)^2 + (y-8)^2 = 4

This equation represents a circle in the standard form, which is:

(x - h)^2 + (y - k)^2 = r^2

Where:

• (h, k) represents the center of the circle.
• r represents the radius of the circle.

Let's break down our equation:

• (x - 1)^2 + (y - 8)^2 = 4

From this equation, we can identify the following:

• Center: (1, 8)
• Radius: √4 = 2

Visualizing the Circle

To visualize this circle, we can follow these steps:

1. Locate the center: Plot the point (1, 8) on a coordinate plane.
2. Draw the radius: From the center, draw a line segment of length 2 units in all directions (up, down, left, right).
3. Complete the circle: Connect the endpoints of these line segments to form a circle.

This circle will be centered at (1, 8) and will have a radius of 2 units.

Key Properties and Applications

This equation and its graphical representation allow us to understand and analyze a circle's properties. Here are some key applications:

• Finding distance: The equation can be used to find the distance between any point on the circle and its center, as this distance will always be equal to the radius.
• Geometric analysis: Understanding the circle's center and radius helps in analyzing geometric relationships involving the circle.
• Real-world applications: Circles appear in various real-world scenarios, such as wheel motion, satellite orbits, and circular structures. This equation provides a mathematical tool to model and analyze these applications.

Conclusion

The equation (x-1)^2 + (y-8)^2 = 4 describes a circle with a center at (1, 8) and a radius of 2 units. This standard form equation provides a powerful tool for understanding and applying the properties of circles in various mathematical and real-world contexts.