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Write the rectangular form of the polar equation in general form.

[tex]\[ r = 6 \tan \theta \][/tex]

Assume that all variables represent positive values.
Enter only the nonzero side of the equation.

[tex]\[ \square = 0 \][/tex]


Sagot :

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]