Find an equation equivalent to [tex]2x + 3y = 6[/tex] in polar coordinates.

A. [tex]r(2 \sin \theta + 3 \cos \theta) = 6[/tex]
B. [tex]2 \cos \theta + 3 \sin \theta = 6r[/tex]
C. [tex]r(2 \cos \theta + 3 \sin \theta) = 6[/tex]
D. [tex]2 \sin \theta + 3 \cos \theta = 6r[/tex]

Please select the best answer from the choices provided:
A, B, C, or D.



Answer :

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