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Exploring the Equation (x-7)^2 + y^2 = 64
The equation (x-7)^2 + y^2 = 64 represents a circle in the Cartesian coordinate system. Let's delve into its key characteristics and understand how it's derived.
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 our equation (x-7)^2 + y^2 = 64 with the standard form, we can extract the following information:
- Center: (h, k) = (7, 0)
- Radius: r = √64 = 8
Therefore, the circle is centered at the point (7, 0) and has a radius of 8 units.
Visual Representation
To visualize the circle, we can plot its center at (7, 0) and draw a circle with a radius of 8 units. This circle will intersect the x-axis at (15, 0) and (-1, 0), and the y-axis at (7, 8) and (7, -8).
Key Features
- Symmetry: The circle is symmetric about both the x-axis and the y-axis.
- Perimeter: The perimeter of the circle is given by 2πr = 16π units.
- Area: The area enclosed by the circle is given by πr^2 = 64π square units.
Applications
The equation of a circle finds numerous applications in various fields:
- Geometry: To describe circular shapes and their properties.
- Physics: To model circular motion and wave propagation.
- Engineering: To design circular structures and components.
- Computer Graphics: To represent and manipulate circular objects in digital environments.
Conclusion
The equation (x-7)^2 + y^2 = 64 provides a concise and powerful way to represent a circle with a specific center and radius. By understanding its key features and applications, we can gain valuable insights into the world of circles and their significance in various disciplines.