To convert the polar equation [tex]\( r = 6 \tan \theta \)[/tex] into its rectangular form, we need to use the relationships between polar and rectangular coordinates.
1. Convert [tex]\( \tan \theta \)[/tex] to Cartesian Coordinates:
We know that:
[tex]\[
\tan \theta = \frac{y}{x}
\][/tex]
2. Substitute [tex]\( \tan \theta \)[/tex]:
The polar equation is:
[tex]\[
r = 6 \tan \theta
\][/tex]
Substituting [tex]\(\tan \theta\)[/tex] with [tex]\(\frac{y}{x}\)[/tex]:
[tex]\[
r = 6 \left( \frac{y}{x} \right)
\][/tex]
3. Convert [tex]\( r \)[/tex] to Cartesian Coordinates:
We know that:
[tex]\[
r = \sqrt{x^2 + y^2}
\][/tex]
Substituting [tex]\(r\)[/tex]:
[tex]\[
\sqrt{x^2 + y^2} = 6 \left( \frac{y}{x} \right)
\][/tex]
4. Eliminate the Square Root:
Square both sides of the equation to eliminate the square root:
[tex]\[
(\sqrt{x^2 + y^2})^2 = \left( 6 \left( \frac{y}{x} \right) \right)^2
\][/tex]
Simplifies to:
[tex]\[
x^2 + y^2 = 36 \frac{y^2}{x^2}
\][/tex]
5. Clear the Fraction by Multiplying by [tex]\( x^2 \)[/tex]:
Multiply every term by [tex]\( x^2 \)[/tex] to clear the denominator:
[tex]\[
x^2 (x^2 + y^2) = 36 y^2
\][/tex]
6. Expand and Simplify:
Distribute [tex]\( x^2 \)[/tex]:
[tex]\[
x^4 + x^2 y^2 = 36 y^2
\][/tex]
Rearrange to get all terms on one side:
[tex]\[
x^4 + x^2 y^2 - 36 y^2 = 0
\][/tex]
The rectangular form of the given polar equation, in its general form, is:
[tex]\[
x^4 + x^2 y^2 - 36 y^2 = 0
\][/tex]
