%5E2%2B(y-3)%5E2%3D16.jpeg "(x+1)^2+(y-3)^2=16")
Exploring the Equation: (x+1)^2 + (y-3)^2 = 16
This equation represents a circle in the Cartesian coordinate system. Let's break down its components and understand how to graph it.
Understanding the Equation
The equation is in standard form for a circle, 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.
In our case:
- (h, k) = (-1, 3) This means the center of the circle is at the point (-1, 3).
- r^2 = 16 Therefore, the radius of the circle is r = 4 (the square root of 16).
Graphing the Circle
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Locate the Center: Plot the point (-1, 3) on the coordinate plane.
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Draw the Radius: From the center, draw a line segment of length 4 units in all directions (up, down, left, right). This will give you four points on the circle.
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Connect the Points: Draw a smooth curve connecting these points to form the circle.
Key Properties of the Circle
- Center: (-1, 3)
- Radius: 4
- Diameter: 8
- Circumference: 8π
- Area: 16π
Applications of Circle Equations
Circle equations are used extensively in various fields, including:
- Geometry: Understanding circles and their properties.
- Physics: Describing the motion of objects in circular paths.
- Engineering: Designing circular structures and objects.
- Computer Graphics: Creating circles and other shapes in digital environments.
Understanding the standard form of a circle equation and its components allows us to analyze and visualize circles effectively, paving the way for further exploration and application of this fundamental geometric concept.