Answer :
Certainly! Let's tackle this problem step-by-step.
The equation of the circle given is:
[tex]\[ (x - 2)^2 + (y - 1)^2 = 64 \][/tex]
### (a) Find the center of the circle.
The general form of a circle's equation is:
[tex]\[ (x - h)^2 + (y - k)^2 = r^2 \][/tex]
where [tex]\((h, k)\)[/tex] is the center of the circle and [tex]\(r\)[/tex] is the radius.
By comparing [tex]\((x - 2)^2 + (y - 1)^2 = 64\)[/tex] with the general form, we see that:
[tex]\[ h = 2 \quad \text{and} \quad k = 1 \][/tex]
Therefore, the center of the circle is:
[tex]\[ \boxed{(2, 1)} \][/tex]
### (b) Find the radius of the circle.
The right-hand side of the equation [tex]\((x - 2)^2 + (y - 1)^2 = 64\)[/tex] represents [tex]\(r^2\)[/tex], where [tex]\(r\)[/tex] is the radius of the circle.
Given [tex]\(64 = r^2\)[/tex], we find [tex]\(r\)[/tex] by taking the square root of 64:
[tex]\[ r = \sqrt{64} = 8 \][/tex]
Therefore, the radius of the circle is:
[tex]\[ \boxed{8} \][/tex]
### (c) Graph the circle.
To graph the circle, you start by plotting the center, which is [tex]\((2, 1)\)[/tex]. Then, using the radius of 8, you draw a circle around the center point reaching a distance of 8 units in all directions.
Here is the step-by-step process for sketching:
1. Plot the center at [tex]\((2, 1)\)[/tex] on a coordinate plane.
2. From the center, measure 8 units up, down, left, and right to mark the extents of the circle.
3. Connect these points with a smooth, round curve to form the circle.
Make sure the circle appears symmetrical about the center.
Below is a rough, conceptual graphical representation:
```
y
|
9 - - O - - - - (10, 1)
|
|
1 - - O (2, 1) - - - 10
|
|
-7 - - O - - - - (0, -7)
|
x
```
Here, the center is at [tex]\((2, 1)\)[/tex], and the outermost points are 8 units away from the center in all directions:
- Leftmost point: [tex]\((2-8, 1) = (-6, 1)\)[/tex]
- Rightmost point: [tex]\((2+8, 1)= (10, 1)\)[/tex]
- Topmost point: [tex]\((2, 1+8)= (2, 9)\)[/tex]
- Bottommost point: [tex]\((2, 1-8)= (2, -7)\)[/tex]
Using a graphing tool, you would draw a clean circle through these points around the center.
The equation of the circle given is:
[tex]\[ (x - 2)^2 + (y - 1)^2 = 64 \][/tex]
### (a) Find the center of the circle.
The general form of a circle's equation is:
[tex]\[ (x - h)^2 + (y - k)^2 = r^2 \][/tex]
where [tex]\((h, k)\)[/tex] is the center of the circle and [tex]\(r\)[/tex] is the radius.
By comparing [tex]\((x - 2)^2 + (y - 1)^2 = 64\)[/tex] with the general form, we see that:
[tex]\[ h = 2 \quad \text{and} \quad k = 1 \][/tex]
Therefore, the center of the circle is:
[tex]\[ \boxed{(2, 1)} \][/tex]
### (b) Find the radius of the circle.
The right-hand side of the equation [tex]\((x - 2)^2 + (y - 1)^2 = 64\)[/tex] represents [tex]\(r^2\)[/tex], where [tex]\(r\)[/tex] is the radius of the circle.
Given [tex]\(64 = r^2\)[/tex], we find [tex]\(r\)[/tex] by taking the square root of 64:
[tex]\[ r = \sqrt{64} = 8 \][/tex]
Therefore, the radius of the circle is:
[tex]\[ \boxed{8} \][/tex]
### (c) Graph the circle.
To graph the circle, you start by plotting the center, which is [tex]\((2, 1)\)[/tex]. Then, using the radius of 8, you draw a circle around the center point reaching a distance of 8 units in all directions.
Here is the step-by-step process for sketching:
1. Plot the center at [tex]\((2, 1)\)[/tex] on a coordinate plane.
2. From the center, measure 8 units up, down, left, and right to mark the extents of the circle.
3. Connect these points with a smooth, round curve to form the circle.
Make sure the circle appears symmetrical about the center.
Below is a rough, conceptual graphical representation:
```
y
|
9 - - O - - - - (10, 1)
|
|
1 - - O (2, 1) - - - 10
|
|
-7 - - O - - - - (0, -7)
|
x
```
Here, the center is at [tex]\((2, 1)\)[/tex], and the outermost points are 8 units away from the center in all directions:
- Leftmost point: [tex]\((2-8, 1) = (-6, 1)\)[/tex]
- Rightmost point: [tex]\((2+8, 1)= (10, 1)\)[/tex]
- Topmost point: [tex]\((2, 1+8)= (2, 9)\)[/tex]
- Bottommost point: [tex]\((2, 1-8)= (2, -7)\)[/tex]
Using a graphing tool, you would draw a clean circle through these points around the center.
