Certainly! Let's tackle the question step by step.
### Part A: Equation of the Circle
To find the equation of a circle, you need to use the standard form of the circle equation:
[tex]\[ (x - h)^2 + (y - k)^2 = r^2 \][/tex]
Here:
- [tex]\((h, k)\)[/tex] represents the center of the circle.
- [tex]\(r\)[/tex] is the radius of the circle.
Given:
- The center of the circle [tex]\((h, k)\)[/tex] is [tex]\((4, -5)\)[/tex].
- The diameter of the circle is 12 units.
To find the radius [tex]\(r\)[/tex]:
[tex]\[ r = \frac{\text{diameter}}{2} = \frac{12}{2} = 6 \][/tex]
Next, we square the radius to find [tex]\(r^2\)[/tex]:
[tex]\[ r^2 = 6^2 = 36 \][/tex]
Now, substituting the center [tex]\((h, k) = (4, -5)\)[/tex] and [tex]\(r^2 = 36\)[/tex] into the standard form of the circle equation:
[tex]\[ (x - 4)^2 + (y - (-5))^2 = 36 \][/tex]
or more simply:
[tex]\[ (x - 4)^2 + (y + 5)^2 = 36 \][/tex]
So, the equation of the circle is:
[tex]\[ (x - 4)^2 + (y + 5)^2 = 36 \][/tex]
### Part B: How to Graph the Circle by Hand
Graphing a circle by hand involves a few systematic steps:
1. Plot the Center: Begin by plotting the center of the circle on the coordinate plane. For this circle, the center is at [tex]\((4, -5)\)[/tex].
2. Measure the Radius: From the center, measure the radius of the circle, which is 6 units, in all four cardinal directions:
- 6 units up: move from [tex]\((4, -5)\)[/tex] to [tex]\((4, 1)\)[/tex].
- 6 units down: move from [tex]\((4, -5)\)[/tex] to [tex]\((4, -11)\)[/tex].
- 6 units to the right: move from [tex]\((4, -5)\)[/tex] to [tex]\((10, -5)\)[/tex].
- 6 units to the left: move from [tex]\((4, -5)\)[/tex] to [tex]\((-2, -5)\)[/tex].
3. Mark these Points: Plot these points [tex]\((4, 1)\)[/tex], [tex]\((4, -11)\)[/tex], [tex]\((10, -5)\)[/tex], and [tex]\((-2, -5)\)[/tex] on the graph.
4. Draw the Circle: Using these points as guides, draw a smooth, round curve connecting these points, forming a circle. This represents all the points that are equidistant (the radius) from the center.
### Summary
In conclusion, for a circle with a center at [tex]\((4, -5)\)[/tex] and a diameter of 12 units:
- The equation is [tex]\((x - 4)^2 + (y + 5)^2 = 36\)[/tex].
- To graph by hand, plot the center, measure and mark the radius in four cardinal directions, then connect these points with a smooth curve to complete the circle.
