## The Equation of a Circle: (x - h)² + (y - k)² = r²

The equation **(x - h)² + (y - k)² = r²** represents a fundamental concept in geometry: **the standard form of the equation of a circle**. This equation provides a powerful tool for understanding and describing circles, enabling us to analyze their properties and solve problems involving their geometry.

### Understanding the Components

**(x - h)² + (y - k)²**: This represents the**distance formula**, calculating the distance between any point (x, y) on the circle and the center of the circle (h, k).**r²**: This represents the**square of the radius**of the circle.

### Key Concepts

**Center:**The point (h, k) is the**center**of the circle. It is the point that is equidistant from all points on the circle.**Radius:**The value 'r' represents the**radius**of the circle. It is the distance from the center to any point on the circle.

### Applications

The standard form equation of a circle is used in various applications:

**Finding the equation of a circle:**Given the center and radius, we can directly substitute these values into the equation to get the circle's equation.**Determining the center and radius:**By rearranging the given equation of a circle, we can identify the center (h, k) and the radius (r).**Analyzing circles:**The equation allows us to determine properties like the diameter, circumference, and area of the circle.**Solving geometric problems:**The equation can be used in solving problems related to intersecting circles, tangent lines, and other geometric concepts.

### Example

Let's consider an example:

**Find the equation of a circle with center (2, 3) and radius 5.**

Using the standard form equation:

- (x - h)² + (y - k)² = r²
- (x - 2)² + (y - 3)² = 5²
**(x - 2)² + (y - 3)² = 25**

Therefore, the equation of the circle is **(x - 2)² + (y - 3)² = 25**.

### Conclusion

The equation **(x - h)² + (y - k)² = r²** provides a concise and powerful representation of a circle. This equation allows us to understand, analyze, and manipulate circles in various mathematical and geometric contexts. It serves as a fundamental tool in solving problems related to circles and their properties.