8.3 Polar coordinates  (Page 4/8)

Rewriting a polar equation in cartesian form

Rewrite the polar equation $r=\frac{3}{1-2\cos \theta }$ as a Cartesian equation.

The goal is to eliminate $\theta$ and $r,$ and introduce $x$ and $y.$ We clear the fraction, and then use substitution. In order to replace $r$ with $x$ and $y,$ we must use the expression $x^2+y^2=r^2.$

$$\begin{array}{llll}r=\frac{3}{1-2\cos \theta }\hfill & \hfill & \hfill & \hfill \\ r(1-2\cos \theta )=3\hfill & \hfill & \hfill & \hfill \\ r\left(1-2\left(\frac{x}{r}\right)\right)=3\hfill & \hfill & \hfill & \text{Use }\cos \theta =\frac{x}{r}\text{ to eliminate }\theta .\hfill \\ r-2x=3\hfill & \hfill & \hfill & \hfill \\ r=3+2x\hfill & \hfill & \hfill & \text{Isolate }r.\hfill \\ r^2={(3+2x)}^2\hfill & \hfill & \hfill & \text{Square both sides}.\hfill \\ x^2+y^2={(3+2x)}^2\hfill & \hfill & \hfill & \text{Use }{x}^2+y^2=r^2.\hfill \end{array}$$

The Cartesian equation is $x^2+y^2={\left(3+2x\right)}^2.$ However, to graph it, especially using a graphing calculator or computer program, we want to isolate $y.$

$$\begin{array}{l}{x}^2+y^2={\left(3+2x\right)}^2\hfill \\ y^2={\left(3+2x\right)}^2-x^2\hfill \\ y=\pm \sqrt{{\left(3+2x\right)}^2-x^2}\hfill \end{array}$$

When our entire equation has been changed from $r$ and $\theta$ to $x$ and $y,$ we can stop, unless asked to solve for $y$ or simplify. See [link] .

The “hour-glass” shape of the graph is called a hyperbola . Hyperbolas have many interesting geometric features and applications, which we will investigate further in Analytic Geometry .

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Key equations

Key concepts

• The polar grid is represented as a series of concentric circles radiating out from the pole, or origin.
• To plot a point in the form $\left(r,\theta \right),\theta >0,$ move in a counterclockwise direction from the polar axis by an angle of $\theta ,$ and then extend a directed line segment from the pole the length of $r$ in the direction of $\theta .$ If $\theta$ is negative, move in a clockwise direction, and extend a directed line segment the length of $r$ in the direction of $\theta .$ See [link] .
• If $r$ is negative, extend the directed line segment in the opposite direction of $\theta .$ See [link] .
• To convert from polar coordinates to rectangular coordinates, use the formulas $x=r\cos \theta$ and $y=r\sin \theta .$ See [link] and [link] .
• To convert from rectangular coordinates to polar coordinates, use one or more of the formulas: $\cos \theta =\frac{x}{r},\sin \theta =\frac{y}{r},\tan \theta =\frac{y}{x},$ and $r=\sqrt{x^2+y^2}.$ See [link] .
• Transforming equations between polar and rectangular forms means making the appropriate substitutions based on the available formulas, together with algebraic manipulations. See [link] , [link] , and [link] .
• Using the appropriate substitutions makes it possible to rewrite a polar equation as a rectangular equation, and then graph it in the rectangular plane. See [link] , [link] , and [link] .

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