## Unveiling the Circle: (x+5)² + (y-6)² = 121

The equation (x+5)² + (y-6)² = 121 represents a circle, a fundamental shape in geometry. Let's explore its properties and understand its meaning.

### The Standard Form of a Circle

The general equation for a circle is:

**(x - h)² + (y - k)² = r²**

Where:

**(h, k)**represents the coordinates of the center of the circle.**r**represents the radius of the circle.

### Analyzing the Equation

Comparing our equation (x+5)² + (y-6)² = 121 to the standard form, we can identify the following:

**Center:**The center of the circle is at (-5, 6), since (h, k) = (-5, 6).**Radius:**The radius of the circle is 11 units, since r² = 121, and r = √121 = 11.

### Visualizing the Circle

To visualize the circle, we can plot its center at (-5, 6) and draw a circle with a radius of 11 units extending outward from the center in all directions.

### Key Properties

**Symmetry:**The circle is symmetrical about both the x-axis and y-axis.**Circumference:**The circumference of the circle is 2πr = 2π(11) = 22π units.**Area:**The area of the circle is πr² = π(11)² = 121π square units.

### Applications

Understanding the equation of a circle has applications in various fields, including:

**Geometry:**Analyzing circles and their relationships with other geometric shapes.**Physics:**Describing circular motion and trajectories.**Computer Graphics:**Representing circular objects in computer graphics and animations.

In conclusion, the equation (x+5)² + (y-6)² = 121 defines a circle with a specific center and radius, offering valuable information about its properties and allowing for its visual representation and analysis.