Question
Rewrite in polar form:x squared plus y squared minus 2 y equals 7.
(1 point)
Responses
r squared equals 2 sine theta plus 7
Image with alt text: r squared equals 2 sine theta plus 7
r squared equals 2 sine theta minus 7
Image with alt text: r squared equals 2 sine theta minus 7
r squared equals 2 r sine theta minus 7
Image with alt text: r squared equals 2 r sine theta minus 7
r squared equals 2 r sine theta plus 7
Rewrite in polar form: x = 12.
(1 point)
Responses
r = 12 cos θ
r = 12 cos θ
r = 12 sin θ
r = 12 sin θ
r = 12 sec θ
r = 12 sec θ
r = 12 csc θ
Rewrite in rectangular form: r = 5.
(1 point)
Responses
x squared plus y squared equals 25
Image with alt text: x squared plus y squared equals 25
x squared plus y squared equals 5
Image with alt text: x squared plus y squared equals 5
x + y = 5
x + y = 5
Start Fraction x squared over 25 End Fraction plus Start Fraction y squared over 25 End Fraction equals 25
Rewrite in rectangular form: r = 6 cos θ.
(1 point)
Responses
left-parenthesis x plus 3 right-parenthesis squared plus y squared equals 9
Image with alt text: left-parenthesis x plus 3 right-parenthesis squared plus y squared equals 9
left-parenthesis x minus 3 right-parenthesis squared plus y squared equals 6
Image with alt text: left-parenthesis x minus 3 right-parenthesis squared plus y squared equals 6
left-parenthesis x minus 3 right-parenthesis squared minus y squared equals 6
Image with alt text: left-parenthesis x minus 3 right-parenthesis squared minus y squared equals 6
left-parenthesis x minus 3 right-parenthesis squared plus y squared equals 9
(1 point)
Responses
r squared equals 2 sine theta plus 7
Image with alt text: r squared equals 2 sine theta plus 7
r squared equals 2 sine theta minus 7
Image with alt text: r squared equals 2 sine theta minus 7
r squared equals 2 r sine theta minus 7
Image with alt text: r squared equals 2 r sine theta minus 7
r squared equals 2 r sine theta plus 7
Rewrite in polar form: x = 12.
(1 point)
Responses
r = 12 cos θ
r = 12 cos θ
r = 12 sin θ
r = 12 sin θ
r = 12 sec θ
r = 12 sec θ
r = 12 csc θ
Rewrite in rectangular form: r = 5.
(1 point)
Responses
x squared plus y squared equals 25
Image with alt text: x squared plus y squared equals 25
x squared plus y squared equals 5
Image with alt text: x squared plus y squared equals 5
x + y = 5
x + y = 5
Start Fraction x squared over 25 End Fraction plus Start Fraction y squared over 25 End Fraction equals 25
Rewrite in rectangular form: r = 6 cos θ.
(1 point)
Responses
left-parenthesis x plus 3 right-parenthesis squared plus y squared equals 9
Image with alt text: left-parenthesis x plus 3 right-parenthesis squared plus y squared equals 9
left-parenthesis x minus 3 right-parenthesis squared plus y squared equals 6
Image with alt text: left-parenthesis x minus 3 right-parenthesis squared plus y squared equals 6
left-parenthesis x minus 3 right-parenthesis squared minus y squared equals 6
Image with alt text: left-parenthesis x minus 3 right-parenthesis squared minus y squared equals 6
left-parenthesis x minus 3 right-parenthesis squared plus y squared equals 9
Answers
GPT-4o mini
Sure! Let's convert each equation step by step and provide the answers based on typical transformations between rectangular and polar coordinates.
1. **Rewrite in polar form: \(x^2 + y^2 - 2y = 7\)**.
First, recall that in polar coordinates:
- \(x = r \cos \theta\)
- \(y = r \sin \theta\)
- \(x^2 + y^2 = r^2\)
We can rewrite the equation as follows:
\[
r^2 - 2(r \sin \theta) = 7
\]
\[
r^2 = 2r \sin \theta + 7
\]
So the correct response is:
**\(r^2 = 2r \sin \theta + 7\)**.
2. **Rewrite in polar form: \(x = 12\)**.
In polar coordinates, \(x\) is represented as \(r \cos \theta\):
\[
r \cos \theta = 12
\]
Solving for \(r\), we get:
\[
r = \frac{12}{\cos \theta} = 12 \sec \theta
\]
Thus, the response is:
**\(r = 12 \sec \theta\)**.
3. **Rewrite in rectangular form: \(r = 5\)**.
The equation \(r = 5\) in rectangular coordinates becomes:
\[
r^2 = 5^2 \implies x^2 + y^2 = 25
\]
Therefore, the correct response is:
**\(x^2 + y^2 = 25\)**.
4. **Rewrite in rectangular form: \(r = 6 \cos \theta\)**.
Using the same polar to rectangular transformation, we have:
\[
r \cos \theta = 6
\]
Substituting \(r = \sqrt{x^2 + y^2}\) and solving gives:
\[
x = 6
\]
Squaring both sides:
\[
x^2 + y^2 = 36
\]
Thus, we can rewrite this equation as:
\[
(x - 3)^2 + y^2 = 9
\]
So the answer is:
**\((x - 3)^2 + y^2 = 9\)**.
In summary:
1. \(r^2 = 2r \sin \theta + 7\)
2. \(r = 12 \sec \theta\)
3. \(x^2 + y^2 = 25\)
4. \((x - 3)^2 + y^2 = 9\)
1. **Rewrite in polar form: \(x^2 + y^2 - 2y = 7\)**.
First, recall that in polar coordinates:
- \(x = r \cos \theta\)
- \(y = r \sin \theta\)
- \(x^2 + y^2 = r^2\)
We can rewrite the equation as follows:
\[
r^2 - 2(r \sin \theta) = 7
\]
\[
r^2 = 2r \sin \theta + 7
\]
So the correct response is:
**\(r^2 = 2r \sin \theta + 7\)**.
2. **Rewrite in polar form: \(x = 12\)**.
In polar coordinates, \(x\) is represented as \(r \cos \theta\):
\[
r \cos \theta = 12
\]
Solving for \(r\), we get:
\[
r = \frac{12}{\cos \theta} = 12 \sec \theta
\]
Thus, the response is:
**\(r = 12 \sec \theta\)**.
3. **Rewrite in rectangular form: \(r = 5\)**.
The equation \(r = 5\) in rectangular coordinates becomes:
\[
r^2 = 5^2 \implies x^2 + y^2 = 25
\]
Therefore, the correct response is:
**\(x^2 + y^2 = 25\)**.
4. **Rewrite in rectangular form: \(r = 6 \cos \theta\)**.
Using the same polar to rectangular transformation, we have:
\[
r \cos \theta = 6
\]
Substituting \(r = \sqrt{x^2 + y^2}\) and solving gives:
\[
x = 6
\]
Squaring both sides:
\[
x^2 + y^2 = 36
\]
Thus, we can rewrite this equation as:
\[
(x - 3)^2 + y^2 = 9
\]
So the answer is:
**\((x - 3)^2 + y^2 = 9\)**.
In summary:
1. \(r^2 = 2r \sin \theta + 7\)
2. \(r = 12 \sec \theta\)
3. \(x^2 + y^2 = 25\)
4. \((x - 3)^2 + y^2 = 9\)