To rewrite the given equation [tex]\( x^2 + y^2 - 2y = 7 \)[/tex] in polar form, we need to follow a structured approach. Let's go through it step-by-step:
1. Recall the polar coordinate definitions:
- [tex]\( x = r \cos \theta \)[/tex]
- [tex]\( y = r \sin \theta \)[/tex]
2. Substitute the polar coordinates into the given Cartesian equation:
- Given: [tex]\( x^2 + y^2 - 2y = 7 \)[/tex]
- Substitute [tex]\( x = r \cos \theta \)[/tex] and [tex]\( y = r \sin \theta \)[/tex]:
[tex]\[
(r \cos \theta)^2 + (r \sin \theta)^2 - 2(r \sin \theta) = 7
\][/tex]
3. Simplify the equation:
- Use the Pythagorean identity [tex]\( \cos^2 \theta + \sin^2 \theta = 1 \)[/tex]:
[tex]\[
r^2 \cos^2 \theta + r^2 \sin^2 \theta - 2r \sin \theta = 7
\][/tex]
- Combine the like terms (since [tex]\( \cos^2 \theta + \sin^2 \theta = 1 \)[/tex]):
[tex]\[
r^2 (\cos^2 \theta + \sin^2 \theta) - 2r \sin \theta = 7
\][/tex]
- Simplify using the identity:
[tex]\[
r^2 (1) - 2r \sin \theta = 7
\][/tex]
[tex]\[
r^2 - 2r \sin \theta = 7
\][/tex]
4. Rearrange the equation to express [tex]\( r^2 \)[/tex] clearly:
[tex]\[
r^2 = 2r \sin \theta + 7
\][/tex]
Hence, the polar form of the given Cartesian equation [tex]\( x^2 + y^2 - 2y = 7 \)[/tex] is:
[tex]\[
r^2 = 2r \sin \theta + 7
\][/tex]
Among the provided options, the correct one is:
- [tex]\( r^2 = 2r \sin \theta + 7 \)[/tex]
So, the index of the correct option is:
4
