(x-1)^2+(y-1)^2=1

3 min read Jun 17, 2024
(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:

  1. Locate the center: Plot the point (1, 1) on the coordinate plane.
  2. Draw the radius: From the center, draw a line segment of length 1 unit in all directions.
  3. 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.

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