Question

Question
Convert the rectangular equation x squared plus y squared minus 8 y equals 0 into a polar equation.
(1 point)
Responses

r = 8 cos θ
r = 8 cos θ

lower r equals start root 8 end root cosine theta
Image with alt text: lower r equals start root 8 end root cosine theta

r = 8 sin θ
r = 8 sin θ

lower r equals start root 8 end root sine theta

Answer 1

To convert the rectangular equation \( x^2 + y^2 - 8y = 0 \) into polar coordinates, we will use the following substitutions:

- \( x = r \cos \theta \)
- \( y = r \sin \theta \)
- \( x^2 + y^2 = r^2 \)

Let's begin by rewriting the equation:

\[
x^2 + y^2 - 8y = 0
\]

Now, substitute the polar coordinate expressions:

\[
r^2 - 8(r \sin \theta) = 0
\]

Next, we can factor out \( r \):

\[
r^2 - 8r \sin \theta = 0
\]

This can be factored further:

\[
r(r - 8 \sin \theta) = 0
\]

Setting each factor to zero gives us two cases:

1. \( r = 0 \) (the origin)
2. \( r - 8 \sin \theta = 0 \) which simplifies to \( r = 8 \sin \theta \)

Thus, the polar equation corresponding to the given rectangular equation is:

\[
r = 8 \sin \theta
\]

So, the correct response is:

**r = 8 sin θ**