Question

Rewrite in polar form: x squared plus y squared minus 6 y minus 8 equals 0.
(1 point)
Responses

r squared equals 6 r plus 6 sine theta plus 8
Image with alt text: r squared equals 6 r plus 6 sine theta plus 8

r squared equals 6 r sine theta plus 8
Image with alt text: r squared equals 6 r sine theta plus 8

r squared equals 6 r plus 6 cosine theta plus 8
Image with alt text: r squared equals 6 r plus 6 cosine theta plus 8

r squared equals 6 r cosine theta plus 8
Image with alt text: r squared equals 6 r cosine theta plus 8

Answer 1

To convert the equation \( x^2 + y^2 - 6y - 8 = 0 \) into polar form, we can rewrite \( x \) and \( y \) in terms of polar coordinates:

\[
x = r \cos \theta \quad \text{and} \quad y = r \sin \theta
\]

Substituting these into the equation gives:

\[
(r \cos \theta)^2 + (r \sin \theta)^2 - 6(r \sin \theta) - 8 = 0
\]

This simplifies as follows:

\[
r^2 \cos^2 \theta + r^2 \sin^2 \theta - 6r \sin \theta - 8 = 0
\]

Since \( \cos^2 \theta + \sin^2 \theta = 1 \), we can combine the terms:

\[
r^2 - 6r \sin \theta - 8 = 0
\]

Rearranging this gives us:

\[
r^2 = 6r \sin \theta + 8
\]

Thus, the polar form of the given equation is:

\[
r^2 = 6r \sin \theta + 8
\]

So, the correct choice from your options is:

**r squared equals 6 r sine theta plus 8**.