To graph the circle given by the equation \((x - 1)^2 + (y - 1)^2 = 81\), follow these step-by-step instructions:
1. Identify the Center and Radius of the Circle:
The equation of the circle is in the standard form \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center of the circle and \(r\) is the radius.
- From the given equation \((x - 1)^2 + (y - 1)^2 = 81\), we can see that:
- The center \((h, k)\) is \((1, 1)\).
- The radius \(r\) is \(\sqrt{81} = 9\).
2. Plot the Center of the Circle:
- Locate the point \((1, 1)\) on the coordinate plane and place a dot or a small mark.
3. Draw the Circle Using the Radius:
- From the center \((1, 1)\), measure a distance of 9 units in all directions (up, down, left, and right).
- Mark points that are 9 units away from the center in each of these directions:
- To the right: \((1 + 9, 1)\) = \((10, 1)\)
- To the left: \((1 - 9, 1)\) = \((-8, 1)\)
- Upwards: \((1, 1 + 9)\) = \((1, 10)\)
- Downwards: \((1, 1 - 9)\) = \((1, -8)\)
4. Draw the Perimeter of the Circle:
- Smoothly connect these points with a round curve to complete the circle. Make sure that all points lying 9 units away from the center \((1, 1)\) are included in this curve.
Here's a summary of the critical points and attributes for graphing the circle:
- Center: \((1, 1)\)
- Radius: \(9\)
- Key perimeter points: \((10, 1)\), \((-8, 1)\), \((1, 10)\), \((1, -8)\)
By following these steps, you will correctly graph the circle given by the equation [tex]\((x - 1)^2 + (y - 1)^2 = 81\)[/tex].
