%5E2%2By%5E2%3D16.jpeg "(x-1)^2+y^2=16")
Exploring the Equation: (x-1)^2 + y^2 = 16
The equation (x-1)^2 + y^2 = 16 represents a circle in the Cartesian coordinate system. Let's delve into understanding this equation and its key properties.
Understanding the Equation
The equation is in the standard form of a circle:
(x - h)^2 + (y - k)^2 = r^2
Where:
- (h, k) is the center of the circle
- r is the radius of the circle
In our equation:
- (h, k) = (1, 0), indicating the center of the circle is at the point (1, 0).
- r^2 = 16, meaning the radius of the circle is r = 4.
Visualizing the Circle
To visualize the circle, plot the center point (1, 0) on the coordinate plane. Then, from this center point, draw a circle with a radius of 4 units.
Key Properties
Here are some key properties of this circle:
- Center: (1, 0)
- Radius: 4
- Diameter: 8
- Circumference: 8π
- Area: 16π
Applications
Understanding circles and their equations is fundamental in various fields, including:
- Geometry: Analyzing geometric shapes and their relationships.
- Trigonometry: Calculating distances and angles using trigonometric functions.
- Physics: Describing circular motion and orbits.
- Engineering: Designing structures, machines, and systems.
Conclusion
The equation (x-1)^2 + y^2 = 16 represents a circle centered at (1, 0) with a radius of 4. This understanding opens doors to exploring various mathematical concepts and applications related to circles.