(x-h)² + (y-k)² = r²

Definition

The equation (x-h)² + (y-k)² = r² represents the standard form of a circle in the Cartesian coordinate plane. Here, (h, k) denotes the center of the circle, while 'r' represents the radius. Understanding this equation helps in analyzing the geometric properties of circles and their placement on a coordinate grid.

5 Must Know Facts For Your Next Test

1. In the equation, if r = 0, it represents a single point located at (h, k).
2. The values h and k can shift the circle around the coordinate plane, affecting its position without altering its size.
3. To graph a circle from this equation, identify the center and measure out 'r' units in all directions to outline the circle's boundary.
4. This equation can be derived from the distance formula, which relates to how far any point on the circle is from its center.
5. Circles are a fundamental component in understanding conic sections, as they serve as one of the basic shapes studied in geometry.

Review Questions

• How does changing the values of h and k affect the position of a circle on a graph?
  • Changing h and k shifts the center of the circle along the x-axis and y-axis respectively. For instance, increasing h moves the circle to the right while decreasing it moves it to the left. Similarly, increasing k shifts the circle upwards and decreasing it shifts it downwards. This demonstrates how adjustments to these values can relocate circles anywhere on a coordinate plane without changing their size.
• Explain how you would derive the equation of a circle given its center and radius.
  • To derive the equation of a circle when given its center at (h, k) and radius 'r', start by applying the distance formula. The distance from any point (x, y) on the circle to the center must equal 'r'. Therefore, we can express this relationship mathematically as √((x-h)² + (y-k)²) = r. Squaring both sides eliminates the square root, leading directly to (x-h)² + (y-k)² = r². This shows how central coordinates and radius work together in defining a circle's shape.
• Evaluate how understanding the standard form of a circle can enhance your comprehension of other geometric concepts like conic sections.
  • Understanding (x-h)² + (y-k)² = r² provides a foundational grasp of circles, which are one type of conic section. This knowledge aids in recognizing how circles relate to ellipses, parabolas, and hyperbolas through their equations and geometric properties. By mastering circles first, you can better analyze and differentiate between these other shapes based on how their equations are structured. This solid base will help you make connections between different geometric forms and their applications.

Related terms

Center of a Circle: The point (h, k) in the circle's equation, which represents the exact midpoint of the circle where all points are equidistant from.

Radius: The distance from the center of the circle to any point on its circumference, denoted by 'r' in the equation.

Circumference: The total distance around the circle, calculated using the formula C = 2πr, where 'r' is the radius.