To find an equivalent equation in polar coordinates for the given Cartesian equation [tex]\(2x + 3y = 6\)[/tex], let's follow the steps of converting Cartesian coordinates to polar coordinates.
1. Identify the given Cartesian equation:
[tex]\[
2x + 3y = 6
\][/tex]
2. Recall the polar coordinate transformations:
[tex]\[
x = r \cos \theta
\][/tex]
[tex]\[
y = r \sin \theta
\][/tex]
3. Substitute [tex]\(x\)[/tex] and [tex]\(y\)[/tex] with their polar equivalents:
[tex]\[
2(r \cos \theta) + 3(r \sin \theta) = 6
\][/tex]
4. Distribute [tex]\(r\)[/tex] in the equation:
[tex]\[
r(2 \cos \theta + 3 \sin \theta) = 6
\][/tex]
5. Simplify the equation:
[tex]\[
r(2 \cos \theta + 3 \sin \theta) = 6
\][/tex]
Now, let us compare the equations given in the options:
- a: [tex]\[r(2 \sin \theta + 3 \cos \theta) = 6\][/tex]
- b: [tex]\[2 \cos \theta + 3 \sin \theta = 6 r\][/tex]
- c: [tex]\[r(2 \cos \theta + 3 \sin \theta) = 6\][/tex]
- d: [tex]\[2 \sin \theta + 3 \cos \theta = 6 r\][/tex]
From our simplification step, we can see that the equivalent polar coordinate equation we derived is:
[tex]\[
r(2 \cos \theta + 3 \sin \theta) = 6
\][/tex]
This matches option (c). Thus, the correct choice is:
C
