Question
Question 1
A)Calculate the center and radius of the circle by completing the square of the equation x2+y2+12x−18y+17=0.(1 point)
Responses
center=(−12,18), radius=20
center= left parenthesis negative 12 comma 18 right parenthesis , radius=20
center=(12,−18), radius=20
center= left parenthesis 12 comma negative 18 right parenthesis , radius=20
center=(6,−9), radius=10
center= left parenthesis 6 comma negative 9 right parenthesis , radius=10
center=(−6,9), radius=10
center= left parenthesis negative 6 comma 9 right parenthesis , radius=10
Question 2
A)Using the equation x2+y2−8x+2y=8, calculate the center and radius of the circle by completing the square.(1 point)
Responses
center=(−4,1), radius=5
center= left parenthesis negative 4 comma 1 right parenthesis , radius=5
center=(2,−8), radius=25
center= left parenthesis 2 comma negative 8 right parenthesis , radius=25
center=(8,−2), radius=25
center= left parenthesis 8 comma negative 2 right parenthesis , radius=25
center=(4,−1), radius=5
center= left parenthesis 4 comma negative 1 right parenthesis , radius=5
Question 3
A)Regina is trying to sketch a circle on graph paper with the equation x2+y2−2x+22y+1=0. Calculate where she would need to graph the center and radius of the circle by completing the square.(1 point)
Responses
center=(1,−11), radius=11
center= left parenthesis 1 comma negative 11 right parenthesis , radius=11
center=(−2,22), radius=41
center= left parenthesis negative 2 comma 22 right parenthesis , radius=41
center=(−22,2), radius=41
center= left parenthesis negative 22 comma 2 right parenthesis , radius=41
center=(−1,11), radius=11
center= left parenthesis negative 1 comma 11 right parenthesis , radius=11
Question 4
A)A circle is represented by the equation x2+y2−16x−6y=−24. Complete the square to discover the center and radius of the circle.(1 point)
Responses
center=(−16,−6), radius=8
center= left parenthesis negative 16 comma negative 6 right parenthesis , radius=8
center=(3,8), radius=7
center= left parenthesis 3 comma 8 right parenthesis , radius=7
center=(8,3), radius=7
center= left parenthesis 8 comma 3 right parenthesis , radius=7
center=(16,6), radius=8
center= left parenthesis 16 comma 6 right parenthesis , radius=8
Question 5
A)Dedric is trying to graph a circle on a computer software program that has a grid. The equation he is graphing is x2+y2−20x−14y+5=0. Where would he place the center of the circle and what is its radius?(1 point)
Responses
center=(20,14), radius=5
center= left parenthesis 20 comma 14 right parenthesis , radius=5
center=(−20,−14), radius=8
center= left parenthesis negative 20 comma negative 14 right parenthesis , radius=8
center=(−10,−7), radius=5
A)Calculate the center and radius of the circle by completing the square of the equation x2+y2+12x−18y+17=0.(1 point)
Responses
center=(−12,18), radius=20
center= left parenthesis negative 12 comma 18 right parenthesis , radius=20
center=(12,−18), radius=20
center= left parenthesis 12 comma negative 18 right parenthesis , radius=20
center=(6,−9), radius=10
center= left parenthesis 6 comma negative 9 right parenthesis , radius=10
center=(−6,9), radius=10
center= left parenthesis negative 6 comma 9 right parenthesis , radius=10
Question 2
A)Using the equation x2+y2−8x+2y=8, calculate the center and radius of the circle by completing the square.(1 point)
Responses
center=(−4,1), radius=5
center= left parenthesis negative 4 comma 1 right parenthesis , radius=5
center=(2,−8), radius=25
center= left parenthesis 2 comma negative 8 right parenthesis , radius=25
center=(8,−2), radius=25
center= left parenthesis 8 comma negative 2 right parenthesis , radius=25
center=(4,−1), radius=5
center= left parenthesis 4 comma negative 1 right parenthesis , radius=5
Question 3
A)Regina is trying to sketch a circle on graph paper with the equation x2+y2−2x+22y+1=0. Calculate where she would need to graph the center and radius of the circle by completing the square.(1 point)
Responses
center=(1,−11), radius=11
center= left parenthesis 1 comma negative 11 right parenthesis , radius=11
center=(−2,22), radius=41
center= left parenthesis negative 2 comma 22 right parenthesis , radius=41
center=(−22,2), radius=41
center= left parenthesis negative 22 comma 2 right parenthesis , radius=41
center=(−1,11), radius=11
center= left parenthesis negative 1 comma 11 right parenthesis , radius=11
Question 4
A)A circle is represented by the equation x2+y2−16x−6y=−24. Complete the square to discover the center and radius of the circle.(1 point)
Responses
center=(−16,−6), radius=8
center= left parenthesis negative 16 comma negative 6 right parenthesis , radius=8
center=(3,8), radius=7
center= left parenthesis 3 comma 8 right parenthesis , radius=7
center=(8,3), radius=7
center= left parenthesis 8 comma 3 right parenthesis , radius=7
center=(16,6), radius=8
center= left parenthesis 16 comma 6 right parenthesis , radius=8
Question 5
A)Dedric is trying to graph a circle on a computer software program that has a grid. The equation he is graphing is x2+y2−20x−14y+5=0. Where would he place the center of the circle and what is its radius?(1 point)
Responses
center=(20,14), radius=5
center= left parenthesis 20 comma 14 right parenthesis , radius=5
center=(−20,−14), radius=8
center= left parenthesis negative 20 comma negative 14 right parenthesis , radius=8
center=(−10,−7), radius=5
Answers
GPT-4o mini
To solve each of the equations for the center and radius of the circles by completing the square, we will go through them one by one.
### Question 1:
Given equation:
\[ x^2 + y^2 + 12x - 18y + 17 = 0 \]
Rearranging:
\[ x^2 + 12x + y^2 - 18y + 17 = 0 \]
\[ x^2 + 12x + y^2 - 18y = -17 \]
**Completing the square:**
- For \(x\):
\[ x^2 + 12x \rightarrow (x + 6)^2 - 36 \]
- For \(y\):
\[ y^2 - 18y \rightarrow (y - 9)^2 - 81 \]
Now substituting back:
\[ (x + 6)^2 - 36 + (y - 9)^2 - 81 = -17 \]
\[ (x + 6)^2 + (y - 9)^2 - 117 = -17 \]
\[ (x + 6)^2 + (y - 9)^2 = 100 \]
**Center and radius:**
- Center: \((-6, 9)\)
- Radius: \(10\)
**Correct Response:**
- center=(−6,9), radius=10
---
### Question 2:
Given equation:
\[ x^2 + y^2 - 8x + 2y = 8 \]
Rearranging:
\[ x^2 - 8x + y^2 + 2y = 8 \]
**Completing the square:**
- For \(x\):
\[ x^2 - 8x \rightarrow (x - 4)^2 - 16 \]
- For \(y\):
\[ y^2 + 2y \rightarrow (y + 1)^2 - 1 \]
Now substituting back:
\[ (x - 4)^2 - 16 + (y + 1)^2 - 1 = 8 \]
\[ (x - 4)^2 + (y + 1)^2 - 17 = 8 \]
\[ (x - 4)^2 + (y + 1)^2 = 25 \]
**Center and radius:**
- Center: \((4, -1)\)
- Radius: \(5\)
**Correct Response:**
- center=(4,−1), radius=5
---
### Question 3:
Given equation:
\[ x^2 + y^2 - 2x + 22y + 1 = 0 \]
Rearranging:
\[ x^2 - 2x + y^2 + 22y + 1 = 0 \]
**Completing the square:**
- For \(x\):
\[ x^2 - 2x \rightarrow (x - 1)^2 - 1 \]
- For \(y\):
\[ y^2 + 22y \rightarrow (y + 11)^2 - 121 \]
Now substituting back:
\[ (x - 1)^2 - 1 + (y + 11)^2 - 121 + 1 = 0 \]
\[ (x - 1)^2 + (y + 11)^2 - 121 = 0 \]
\[ (x - 1)^2 + (y + 11)^2 = 121 \]
**Center and radius:**
- Center: \((1, -11)\)
- Radius: \(11\)
**Correct Response:**
- center=(1,−11), radius=11
---
### Question 4:
Given equation:
\[ x^2 + y^2 - 16x - 6y = -24 \]
Rearranging:
\[ x^2 - 16x + y^2 - 6y = -24 \]
**Completing the square:**
- For \(x\):
\[ x^2 - 16x \rightarrow (x - 8)^2 - 64 \]
- For \(y\):
\[ y^2 - 6y \rightarrow (y - 3)^2 - 9 \]
Now substituting back:
\[ (x - 8)^2 - 64 + (y - 3)^2 - 9 = -24 \]
\[ (x - 8)^2 + (y - 3)^2 - 73 = -24 \]
\[ (x - 8)^2 + (y - 3)^2 = 49 \]
**Center and radius:**
- Center: \((8, 3)\)
- Radius: \(7\)
**Correct Response:**
- center=(8,3), radius=7
---
### Question 5:
Given equation:
\[ x^2 + y^2 - 20x - 14y + 5 = 0 \]
Rearranging:
\[ x^2 - 20x + y^2 - 14y + 5 = 0 \]
**Completing the square:**
- For \(x\):
\[ x^2 - 20x \rightarrow (x - 10)^2 - 100 \]
- For \(y\):
\[ y^2 - 14y \rightarrow (y - 7)^2 - 49 \]
Now substituting back:
\[ (x - 10)^2 - 100 + (y - 7)^2 - 49 + 5 = 0 \]
\[ (x - 10)^2 + (y - 7)^2 - 144 = 0 \]
\[ (x - 10)^2 + (y - 7)^2 = 144 \]
**Center and radius:**
- Center: \((10, 7)\)
- Radius: \(12\)
**Note:** There might be a miscalculation here as none of the provided options correspond accurately to the findings.
None of the responses for question 5 are correct based on my calculations.
---
To summarize the correct results:
1. Question 1: center=(-6,9), radius=10
2. Question 2: center=(4,−1), radius=5
3. Question 3: center=(1,−11), radius=11
4. Question 4: center=(8,3), radius=7
5. Question 5: No correct responses available based on calculations.
### Question 1:
Given equation:
\[ x^2 + y^2 + 12x - 18y + 17 = 0 \]
Rearranging:
\[ x^2 + 12x + y^2 - 18y + 17 = 0 \]
\[ x^2 + 12x + y^2 - 18y = -17 \]
**Completing the square:**
- For \(x\):
\[ x^2 + 12x \rightarrow (x + 6)^2 - 36 \]
- For \(y\):
\[ y^2 - 18y \rightarrow (y - 9)^2 - 81 \]
Now substituting back:
\[ (x + 6)^2 - 36 + (y - 9)^2 - 81 = -17 \]
\[ (x + 6)^2 + (y - 9)^2 - 117 = -17 \]
\[ (x + 6)^2 + (y - 9)^2 = 100 \]
**Center and radius:**
- Center: \((-6, 9)\)
- Radius: \(10\)
**Correct Response:**
- center=(−6,9), radius=10
---
### Question 2:
Given equation:
\[ x^2 + y^2 - 8x + 2y = 8 \]
Rearranging:
\[ x^2 - 8x + y^2 + 2y = 8 \]
**Completing the square:**
- For \(x\):
\[ x^2 - 8x \rightarrow (x - 4)^2 - 16 \]
- For \(y\):
\[ y^2 + 2y \rightarrow (y + 1)^2 - 1 \]
Now substituting back:
\[ (x - 4)^2 - 16 + (y + 1)^2 - 1 = 8 \]
\[ (x - 4)^2 + (y + 1)^2 - 17 = 8 \]
\[ (x - 4)^2 + (y + 1)^2 = 25 \]
**Center and radius:**
- Center: \((4, -1)\)
- Radius: \(5\)
**Correct Response:**
- center=(4,−1), radius=5
---
### Question 3:
Given equation:
\[ x^2 + y^2 - 2x + 22y + 1 = 0 \]
Rearranging:
\[ x^2 - 2x + y^2 + 22y + 1 = 0 \]
**Completing the square:**
- For \(x\):
\[ x^2 - 2x \rightarrow (x - 1)^2 - 1 \]
- For \(y\):
\[ y^2 + 22y \rightarrow (y + 11)^2 - 121 \]
Now substituting back:
\[ (x - 1)^2 - 1 + (y + 11)^2 - 121 + 1 = 0 \]
\[ (x - 1)^2 + (y + 11)^2 - 121 = 0 \]
\[ (x - 1)^2 + (y + 11)^2 = 121 \]
**Center and radius:**
- Center: \((1, -11)\)
- Radius: \(11\)
**Correct Response:**
- center=(1,−11), radius=11
---
### Question 4:
Given equation:
\[ x^2 + y^2 - 16x - 6y = -24 \]
Rearranging:
\[ x^2 - 16x + y^2 - 6y = -24 \]
**Completing the square:**
- For \(x\):
\[ x^2 - 16x \rightarrow (x - 8)^2 - 64 \]
- For \(y\):
\[ y^2 - 6y \rightarrow (y - 3)^2 - 9 \]
Now substituting back:
\[ (x - 8)^2 - 64 + (y - 3)^2 - 9 = -24 \]
\[ (x - 8)^2 + (y - 3)^2 - 73 = -24 \]
\[ (x - 8)^2 + (y - 3)^2 = 49 \]
**Center and radius:**
- Center: \((8, 3)\)
- Radius: \(7\)
**Correct Response:**
- center=(8,3), radius=7
---
### Question 5:
Given equation:
\[ x^2 + y^2 - 20x - 14y + 5 = 0 \]
Rearranging:
\[ x^2 - 20x + y^2 - 14y + 5 = 0 \]
**Completing the square:**
- For \(x\):
\[ x^2 - 20x \rightarrow (x - 10)^2 - 100 \]
- For \(y\):
\[ y^2 - 14y \rightarrow (y - 7)^2 - 49 \]
Now substituting back:
\[ (x - 10)^2 - 100 + (y - 7)^2 - 49 + 5 = 0 \]
\[ (x - 10)^2 + (y - 7)^2 - 144 = 0 \]
\[ (x - 10)^2 + (y - 7)^2 = 144 \]
**Center and radius:**
- Center: \((10, 7)\)
- Radius: \(12\)
**Note:** There might be a miscalculation here as none of the provided options correspond accurately to the findings.
None of the responses for question 5 are correct based on my calculations.
---
To summarize the correct results:
1. Question 1: center=(-6,9), radius=10
2. Question 2: center=(4,−1), radius=5
3. Question 3: center=(1,−11), radius=11
4. Question 4: center=(8,3), radius=7
5. Question 5: No correct responses available based on calculations.