%5E2%2By%5E2%3D16.jpeg "(x-2)^2+y^2=16")
Exploring the Equation: (x-2)^2 + y^2 = 16
The equation (x-2)^2 + y^2 = 16 represents a circle in the Cartesian coordinate system. Let's delve into the details of this equation and what it tells us about the circle.
Understanding the Standard Form
This equation is in the standard form of a circle's equation:
(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 Circle's Properties
By comparing our equation (x-2)^2 + y^2 = 16 to the standard form, we can identify the following:
- Center: The center of the circle is (2, 0). This is because the equation is in the form (x - 2)^2, indicating a horizontal shift of 2 units to the right from the origin.
- Radius: The radius of the circle is 4. This is because the right-hand side of the equation is 16, which is the square of 4 (r^2 = 16).
Visualizing the Circle
Based on the center and radius, we can visualize the circle:
- Plot the center point (2, 0) on the coordinate plane.
- From the center, move 4 units in every direction (up, down, left, right) to mark points on the circle's circumference.
- Connect these points to form a complete circle.
Conclusion
The equation (x-2)^2 + y^2 = 16 defines a circle with a center at (2, 0) and a radius of 4. Understanding the standard form of a circle's equation allows us to quickly identify its center and radius, providing a clear understanding of its position and size in the coordinate plane.