To convert the given polar equation [tex]\( r = 3 \cos \theta - \sin \theta \)[/tex] to a rectangular equation, follow these steps:
1. Identify the polar to rectangular transformation identities:
- [tex]\( x = r \cos \theta \)[/tex]
- [tex]\( y = r \sin \theta \)[/tex]
- [tex]\( r^2 = x^2 + y^2 \)[/tex]
2. Multiply both sides of the given polar equation by [tex]\( r \)[/tex]:
[tex]\[
r \cdot r = r \cdot (3 \cos \theta - \sin \theta)
\][/tex]
Which simplifies to:
[tex]\[
r^2 = r \cdot 3 \cos \theta - r \cdot \sin \theta
\][/tex]
3. Use the polar to rectangular transformation identities:
- [tex]\( r^2 = x^2 + y^2 \)[/tex]
- [tex]\( r \cos \theta = x \)[/tex]
- [tex]\( r \sin \theta = y \)[/tex]
Substitute these into the equation:
[tex]\[
x^2 + y^2 = 3x - y
\][/tex]
4. Rearrange the equation to get all terms on one side:
[tex]\[
x^2 + y^2 - 3x + y = 0
\][/tex]
Therefore, the rectangular form of the given polar equation is:
[tex]\[
x^2 - 3 x + y^2 + y = 0
\][/tex]
So, the correct answer is:
c. [tex]\( x^2 - 3 x + y^2 + y = 0 \)[/tex]
