## Calculus (3rd Edition)

$$3x^2 -2x +4y^2=1.$$
We have $$r=\frac{1}{2-\cos\theta}\Longrightarrow 2r-r\cos\theta =1.$$ Recall that $r\cos \theta=x$ and $r^2=x^2+y^2$: Hence, we get $$2\sqrt{x^2+y^2}-x =1\Longrightarrow 4(x^2+y^2)=(1+x)^2=1+2x+x^2.$$ Finally, we get $$3x^2 -2x +4y^2=1.$$
The passage contains a mathematical derivation that starts with a polar equation and transforms it into a Cartesian equation. No explicit question is asked, and no answer to a question is provided.
no_question
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
## Calculus (3rd Edition)

$$3x^2 -2x +4y^2=1.$$
We have $$r=\frac{1}{2-\cos\theta}\Longrightarrow 2r-r\cos\theta =1.$$ Recall that $r\cos \theta=x$ and $r^2=x^2+y^2$: Hence, we get $$2\sqrt{x^2+y^2}-x =1\Longrightarrow 4(x^2+y^2)=(1+x)^2=1+2x+x^2.$$ Finally, we get $$3x^2 -2x +4y^2=1.$$
user
The passage contains a mathematical derivation that starts with a polar equation and transforms it into a Cartesian equation. No explicit question is asked, and no answer to a question is provided.

TYPE: NO_QUESTION