Polar to Rectangular Oh My

[MAC-A-TAC]

Hello,

I need help converting r = 4/1-2cos theta to rectangular coordinates. If possible, please show me each step of the conversion.
I am preparing for my final exam and this study problem has me perplexed. :?

Thank you in advance for your help.

Mac out.

 

[soroban]

Hello, MAC-A-TAC!

\(\displaystyle \text{Convert to rectangular coordinates:}\:r \:=\:\frac{4}{1-2\cos\theta}\)

\(\displaystyle \text{Multiply by }(1-2\cos\theta)\!:\quad r(1 - 2\cos\theta) \:=\:4\)

\(\displaystyle \text{We have: }\;\;r - 2r\cos\theta \:=\:4\)


\(\displaystyle \text{Then: }\;\;\underbrace{r}_{\sqrt{x^2+y^2}} \:=\:2\underbrace{r\cos\theta}_{x} + 4\quad\Rightarrow\quad \sqrt{x^2+y^2} \:=\:2x + 4\)

\(\displaystyle \text{Square both sides: }\;\;(\sqrt{x^2+y^2})^2 \;=\;(2x+4)^2 \quad\Rightarrow\quad x^2 + y^2 \;=\;4x^2 + 16x + 16\)


\(\displaystyle \text{Therefore: }\;\;y^2 - 3x^2 - 16x \:=\:16 \quad\hdots\;\text{ hyperbola}\)

 

[MAC-A-TAC]

Thank you!