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Exploring the Equation: (x - 1)^2 + y^2 = 4
The equation (x - 1)^2 + y^2 = 4 represents a circle in the standard form. Let's break down its key features and explore its properties.
Understanding the Standard Form
The standard form of a circle's equation 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 equation, (x - 1)^2 + y^2 = 4, we can see:
- h = 1
- k = 0
- r^2 = 4, therefore r = 2
Interpreting the Equation
This means our circle has:
- Center: (1, 0)
- Radius: 2
Visualizing the Circle
To visualize the circle, we can plot the center point (1, 0) on a coordinate plane. Then, draw a circle with a radius of 2 units around this center point.
Key Features
Here are some important features of the circle represented by (x - 1)^2 + y^2 = 4:
- Symmetry: The circle is symmetrical about both the x-axis and the y-axis.
- Area: The area of the circle is calculated as πr^2, which in this case is 4π square units.
- Circumference: The circumference of the circle is calculated as 2πr, which in this case is 4π units.
Applications
Understanding the equation of a circle and its properties is crucial in various fields, including:
- Geometry: Describing and analyzing geometric shapes.
- Physics: Modeling circular motion and forces.
- Computer Graphics: Creating and manipulating circular objects.
- Engineering: Designing circular structures and components.
Conclusion
The equation (x - 1)^2 + y^2 = 4 provides a concise representation of a circle with specific properties. By understanding the standard form of the equation, we can easily determine the center, radius, and other important features of the circle. This knowledge has wide-ranging applications in various fields, making it a fundamental concept in mathematics and beyond.