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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.