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Exploring the Equation (x-3)^2 + y^2 = 4
The equation (x-3)^2 + y^2 = 4 represents a circle in the Cartesian coordinate system. Let's break down its components and explore its properties.
Understanding the Equation
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Standard Form: The equation is in the standard form of a circle: (x-h)^2 + (y-k)^2 = r^2, where:
- (h, k) represents the center of the circle.
- r represents the radius of the circle.
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Center: In our equation, (h, k) = (3, 0), indicating that the center of the circle lies at the point (3, 0).
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Radius: The equation shows r^2 = 4, therefore r = 2. This means the circle has a radius of 2 units.
Visualizing the Circle
To visualize the circle, we can follow these steps:
- Locate the Center: Plot the point (3, 0) on the coordinate plane.
- Draw the Radius: From the center, draw a line segment of length 2 units in all directions (up, down, left, and right).
- Complete the Circle: Connect the endpoints of the radius segments to form a smooth curve, creating the circle.
Properties of the Circle
The equation (x-3)^2 + y^2 = 4 defines a circle with the following properties:
- Center: (3, 0)
- Radius: 2 units
- Diameter: 4 units
- Circumference: 2πr = 4π units
- Area: πr^2 = 4π square units
Applications
Understanding the equation of a circle has various applications in:
- Geometry: Determining the properties of circles and solving problems related to their area, circumference, and other geometric concepts.
- Physics: Modeling the path of objects moving in circular orbits or representing circular objects like planets or satellites.
- Engineering: Designing circular structures, calculating the stress distribution in circular components, and analyzing the behavior of circular objects under various forces.
By understanding the equation and its properties, we can effectively analyze and apply this mathematical representation of a circle in various contexts.