(x-7)2+y2=64

3 min read Jun 17, 2024
(x-7)2+y2=64

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.

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