To convert the given rectangular equation [tex]\(3x^2 + 2y^2 = 18\)[/tex] to polar form, we need to use the relationships between rectangular coordinates [tex]\((x, y)\)[/tex] and polar coordinates [tex]\((r, \theta)\)[/tex]. Specifically, use the substitutions:
[tex]\[ x = r \cos(\theta) \][/tex]
[tex]\[ y = r \sin(\theta) \][/tex]
Let’s substitute [tex]\(x = r \cos(\theta)\)[/tex] and [tex]\(y = r \sin(\theta)\)[/tex] into the given equation:
[tex]\[ 3x^2 + 2y^2 = 18 \][/tex]
Substituting the expressions for [tex]\(x\)[/tex] and [tex]\(y\)[/tex], we get:
[tex]\[ 3 (r \cos(\theta))^2 + 2 (r \sin(\theta))^2 = 18 \][/tex]
Now, let's simplify this expression step by step:
1. Substitute [tex]\(x\)[/tex] and [tex]\(y\)[/tex] with their polar forms:
[tex]\[ 3 (r \cos(\theta))^2 + 2 (r \sin(\theta))^2 = 18 \][/tex]
2. Simplify each term:
[tex]\[ 3 (r^2 \cos^2(\theta)) + 2 (r^2 \sin^2(\theta)) = 18 \][/tex]
3. Factor out [tex]\(r^2\)[/tex]:
[tex]\[ r^2 (3 \cos^2(\theta) + 2 \sin^2(\theta)) = 18 \][/tex]
So, the correct first step(s) to convert the equation into polar form corresponds to option (a):
[tex]\[ 3(r \cos(\theta))^2 + 2 (r \sin(\theta))^2 = 18 \][/tex]
Therefore, the correct answer is:
a. [tex]\(3(r \cos(\theta))^2 + 2 (r \sin(\theta))^2 = 18\)[/tex]
