%5E2%2B(y%2B1)%5E2%3D16.jpeg "(x-2)^2+(y+1)^2=16")
Understanding the Equation: (x-2)^2 + (y+1)^2 = 16
This equation represents a circle in the standard form. Let's break down the components and explore its properties.
Standard Form of a Circle
The standard form of a circle's equation is:
(x - h)^2 + (y - k)^2 = r^2
Where:
- (h, k) represents the coordinates of the circle's center.
- r represents the radius of the circle.
Analyzing the Given Equation
In our equation, (x - 2)^2 + (y + 1)^2 = 16:
- (h, k) = (2, -1), indicating the center of the circle is at the point (2, -1).
- r^2 = 16, meaning the radius is the square root of 16, which is r = 4.
Properties of the Circle
Knowing the center and radius allows us to understand the circle's properties:
- Center: The point (2, -1) is the center of the circle.
- Radius: The distance from the center to any point on the circle is 4 units.
- Circumference: The distance around the circle can be calculated using the formula C = 2πr, which gives us C = 8π.
- Area: The space enclosed by the circle can be calculated using the formula A = πr^2, which gives us A = 16π.
Graphing the Circle
To visualize the circle, we can plot its center (2, -1) on a coordinate plane. Then, using the radius of 4, we can draw a circle around the center point.
In summary, the equation (x - 2)^2 + (y + 1)^2 = 16 describes a circle with a center at (2, -1) and a radius of 4. This understanding allows us to determine its key properties and visualize its position on a coordinate plane.