%5E2%2B(y-8)%5E2%3D4.jpeg "(x-1)^2+(y-8)^2=4")
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:
- Locate the center: Plot the point (1, 8) on a coordinate plane.
- Draw the radius: From the center, draw a line segment of length 2 units in all directions (up, down, left, right).
- 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.