Question
Determine the equations of EACH circle from both suspects and compare them to the original equation in the space provided.
The Circle Conspiracy
(x-2^2 + (y+1)^2 = 36
Suspects A said the circle was centered at (3,4) with the radius of five
Suspects B said the circle was centered at (0,0) with the radius of 7
im arguing that suspect B did it
The Circle Conspiracy
(x-2^2 + (y+1)^2 = 36
Suspects A said the circle was centered at (3,4) with the radius of five
Suspects B said the circle was centered at (0,0) with the radius of 7
im arguing that suspect B did it
Answers
Answered by
GPT-4o mini
To determine the equations of each circle based on the suspects' claims and compare them to the original equation, let's start with the original equation given:
\[
(x - 2)^2 + (y + 1)^2 = 36
\]
This equation 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.
### Extracting Information from Original Equation
From the original equation:
1. The center \((h, k)\) is \((2, -1)\).
2. The radius \(r\) is given by \(r^2 = 36\), so \(r = \sqrt{36} = 6\).
### Analyzing Suspect A's Claim
**Suspect A claims**:
- Center: \((3, 4)\)
- Radius: \(5\)
- To write the equation of the circle based on Suspect A's claim:
The center is \((h, k) = (3, 4)\) and radius \(r = 5\).
Using the standard form:
\[
(x - 3)^2 + (y - 4)^2 = 5^2
\]
\[
(x - 3)^2 + (y - 4)^2 = 25
\]
### Analyzing Suspect B's Claim
**Suspect B claims**:
- Center: \((0, 0)\)
- Radius: \(7\)
- To write the equation of the circle based on Suspect B's claim:
The center is \((h, k) = (0, 0)\) and radius \(r = 7\).
Using the standard form:
\[
(x - 0)^2 + (y - 0)^2 = 7^2
\]
\[
x^2 + y^2 = 49
\]
### Summary of Circle Equations
1. **Original Circle**: \((x - 2)^2 + (y + 1)^2 = 36\) (Center: \((2, -1)\), Radius: \(6\))
2. **Suspect A's Circle**: \((x - 3)^2 + (y - 4)^2 = 25\) (Center: \((3, 4)\), Radius: \(5\))
3. **Suspect B's Circle**: \(x^2 + y^2 = 49\) (Center: \((0, 0)\), Radius: \(7\))
### Comparison
- The center and radius from **Suspect A** do not match the original circle.
- The center and radius from **Suspect B** also do not match the original circle.
### Conclusion
- **Your argument** that suspect B did it is based on the incorrect claim, as neither suspect provided accurate information regarding the center or radius of the original circle, and thus neither suspect is actually representing the original circle's properties.
\[
(x - 2)^2 + (y + 1)^2 = 36
\]
This equation 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.
### Extracting Information from Original Equation
From the original equation:
1. The center \((h, k)\) is \((2, -1)\).
2. The radius \(r\) is given by \(r^2 = 36\), so \(r = \sqrt{36} = 6\).
### Analyzing Suspect A's Claim
**Suspect A claims**:
- Center: \((3, 4)\)
- Radius: \(5\)
- To write the equation of the circle based on Suspect A's claim:
The center is \((h, k) = (3, 4)\) and radius \(r = 5\).
Using the standard form:
\[
(x - 3)^2 + (y - 4)^2 = 5^2
\]
\[
(x - 3)^2 + (y - 4)^2 = 25
\]
### Analyzing Suspect B's Claim
**Suspect B claims**:
- Center: \((0, 0)\)
- Radius: \(7\)
- To write the equation of the circle based on Suspect B's claim:
The center is \((h, k) = (0, 0)\) and radius \(r = 7\).
Using the standard form:
\[
(x - 0)^2 + (y - 0)^2 = 7^2
\]
\[
x^2 + y^2 = 49
\]
### Summary of Circle Equations
1. **Original Circle**: \((x - 2)^2 + (y + 1)^2 = 36\) (Center: \((2, -1)\), Radius: \(6\))
2. **Suspect A's Circle**: \((x - 3)^2 + (y - 4)^2 = 25\) (Center: \((3, 4)\), Radius: \(5\))
3. **Suspect B's Circle**: \(x^2 + y^2 = 49\) (Center: \((0, 0)\), Radius: \(7\))
### Comparison
- The center and radius from **Suspect A** do not match the original circle.
- The center and radius from **Suspect B** also do not match the original circle.
### Conclusion
- **Your argument** that suspect B did it is based on the incorrect claim, as neither suspect provided accurate information regarding the center or radius of the original circle, and thus neither suspect is actually representing the original circle's properties.