Question
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
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
Answered by
GPT-4o mini
To rewrite \( r = 5 \) in rectangular form, we start with the equation of a circle in polar coordinates. The equation is \( r = 5 \), which represents a circle of radius 5 centered at the origin.
In rectangular coordinates, this is given by the equation:
\[
x^2 + y^2 = 5^2 = 25
\]
Thus, the correct response is:
**x squared plus y squared equals 25.**
---
Next, to rewrite \( r = 6 \cos \theta \) in rectangular form, we use the relationships \( r = \sqrt{x^2 + y^2} \) and \( \cos \theta = \frac{x}{r} \).
Starting from the given equation:
\[
r = 6 \cos \theta
\]
Substituting the relationships:
\[
r = 6 \left( \frac{x}{r} \right)
\]
Multiplying both sides by \( r \):
\[
r^2 = 6x
\]
Now substituting \( r^2 = x^2 + y^2 \):
\[
x^2 + y^2 = 6x
\]
Rearranging gives:
\[
x^2 - 6x + y^2 = 0
\]
To write this in standard form, we complete the square for the \( x \) term:
\[
(x - 3)^2 + y^2 = 9
\]
Thus, the correct response is:
**left-parenthesis x minus 3 right-parenthesis squared plus y squared equals 9.**
In rectangular coordinates, this is given by the equation:
\[
x^2 + y^2 = 5^2 = 25
\]
Thus, the correct response is:
**x squared plus y squared equals 25.**
---
Next, to rewrite \( r = 6 \cos \theta \) in rectangular form, we use the relationships \( r = \sqrt{x^2 + y^2} \) and \( \cos \theta = \frac{x}{r} \).
Starting from the given equation:
\[
r = 6 \cos \theta
\]
Substituting the relationships:
\[
r = 6 \left( \frac{x}{r} \right)
\]
Multiplying both sides by \( r \):
\[
r^2 = 6x
\]
Now substituting \( r^2 = x^2 + y^2 \):
\[
x^2 + y^2 = 6x
\]
Rearranging gives:
\[
x^2 - 6x + y^2 = 0
\]
To write this in standard form, we complete the square for the \( x \) term:
\[
(x - 3)^2 + y^2 = 9
\]
Thus, the correct response is:
**left-parenthesis x minus 3 right-parenthesis squared plus y squared equals 9.**