2%2B(y-6)2%3D81-graph.jpeg "(x-5)2+(y-6)2=81 Graph")
The Circle Defined by (x-5)² + (y-6)² = 81
The equation (x-5)² + (y-6)² = 81 represents a circle in the coordinate plane. Let's explore how to graph it and understand its key properties.
Understanding the Equation
The equation is in standard form for a circle: (x - h)² + (y - k)² = r²
- (h, k) represents the center of the circle.
- r represents the radius of the circle.
In our equation, h = 5 and k = 6, indicating the center of the circle is at the point (5, 6). Furthermore, r² = 81, meaning r = 9 (since the radius is always positive).
Graphing the Circle
- Plot the Center: Begin by plotting the point (5, 6) on your coordinate plane.
- Mark the Radius: From the center point, move 9 units to the right, left, up, and down. These points will be on the circle.
- Connect the Points: Draw a smooth curve connecting the points you marked to form the complete circle.
Key Properties
- Center: (5, 6)
- Radius: 9
- Diameter: 18 (twice the radius)
- Circumference: 18π (2πr)
- Area: 81π (πr²)
Example:
Let's consider a point on the circle, say (14, 6). Notice that it is 9 units to the right of the center (5, 6). This confirms that it lies on the circle, as the distance from any point on the circle to its center is always equal to the radius.
In summary, the equation (x-5)² + (y-6)² = 81 defines a circle with a center at (5, 6) and a radius of 9 units. By understanding the standard form of the equation and its components, you can readily graph and analyze circles.