2%2By2%3D81.jpeg "(x+8)2+y2=81")
Exploring the Equation: (x + 8)^2 + y^2 = 81
The equation (x + 8)^2 + y^2 = 81 represents a circle in the coordinate plane. Let's break down how to understand this equation and its properties.
Understanding the Standard Form
The equation is in the standard form of a circle:
(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
In our equation, (x + 8)^2 + y^2 = 81, we can rewrite it as:
(x - (-8))^2 + (y - 0)^2 = 9^2
Now, we can easily identify the center and radius:
- Center: (-8, 0)
- Radius: 9
Visualizing the Circle
With the center and radius, we can visualize the circle:
- Plot the center: (-8, 0) on the coordinate plane.
- Draw a circle: With the center as the origin, draw a circle with a radius of 9 units.
Key Properties
- Symmetry: The circle is symmetrical about both the x-axis and y-axis.
- Diameter: The diameter of the circle is twice the radius, which is 18 units.
- Circumference: The circumference of the circle is calculated using the formula C = 2πr, which is 18π units.
- Area: The area of the circle is calculated using the formula A = πr^2, which is 81π square units.
Conclusion
The equation (x + 8)^2 + y^2 = 81 describes a circle with a center at (-8, 0) and a radius of 9 units. Understanding the standard form and identifying the center and radius allows us to easily visualize and analyze the properties of this circle.