## 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.