%5E2%2B(y-1)%5E2%3D1.jpeg "(x-1)^2+(y-1)^2=1")
Unveiling the Circle: Exploring the Equation (x-1)² + (y-1)² = 1
The equation (x-1)² + (y-1)² = 1 holds a hidden beauty – it represents a circle in the coordinate plane. Let's delve into the details of this equation and understand its significance.
Understanding the Equation
This equation is a standard form of the equation of a circle, which can be expressed as:
(x - h)² + (y - k)² = r²
Where:
- (h, k) represents the center of the circle
- r represents the radius of the circle
In our given equation, (x-1)² + (y-1)² = 1, we can see that:
- h = 1 and k = 1, indicating the center of the circle is at (1, 1).
- r² = 1, implying that the radius of the circle is r = 1.
Visualizing the Circle
To visualize the circle, follow these steps:
- Locate the center: Plot the point (1, 1) on the coordinate plane.
- Draw the radius: From the center, draw a line segment of length 1 unit in all directions.
- Connect the points: Connect the endpoints of these line segments to form a smooth curve.
The resulting curve represents the circle defined by the equation (x-1)² + (y-1)² = 1.
Key Properties of the Circle
- Center: (1, 1)
- Radius: 1
- Circumference: 2πr = 2π
- Area: πr² = π
Applications
The equation of a circle finds widespread applications in various fields, including:
- Geometry: Analyzing geometric shapes and their properties.
- Physics: Describing the motion of objects in a circular path.
- Engineering: Designing circular structures and components.
- Computer graphics: Creating and manipulating circular objects in digital media.
Conclusion
The simple equation (x-1)² + (y-1)² = 1 encapsulates a powerful geometric concept – the circle. Understanding its components and properties enables us to analyze and apply this fundamental shape in various scientific and engineering disciplines.