%5E2%2B(y%2B1)%5E2%3D9.jpeg "(x-5)^2+(y+1)^2=9")
Exploring the Equation: (x - 5)^2 + (y + 1)^2 = 9
The equation (x - 5)^2 + (y + 1)^2 = 9 represents a circle in the Cartesian coordinate system. Let's break down why and how to understand its features:
Understanding the Standard Form
The equation is in the standard form of a circle's equation:
(x - h)^2 + (y - k)^2 = r^2
Where:
- (h, k) represents the center of the circle.
- r represents the radius of the circle.
Identifying the Center and Radius
By comparing the given equation to the standard form, we can directly identify the center and radius:
- Center: (h, k) = (5, -1)
- Radius: r^2 = 9, therefore r = 3
Visualizing the Circle
With the center and radius identified, we can visualize the circle on a graph:
- Plot the center: Locate the point (5, -1) on the coordinate plane.
- Draw the circle: From the center, measure 3 units in all directions (up, down, left, right). Connect these points to form a circle with a radius of 3.
Key Features
The equation (x - 5)^2 + (y + 1)^2 = 9 describes a circle with the following key features:
- Center: (5, -1)
- Radius: 3
- Circumference: 2πr = 6π
- Area: πr^2 = 9π
Conclusion
By understanding the standard form of a circle's equation, we can easily identify the center and radius. This information allows us to accurately visualize and analyze the circle's properties. The equation (x - 5)^2 + (y + 1)^2 = 9 represents a specific circle with a center at (5, -1) and a radius of 3.