# To calculate: The sections represented by the polar equation displaystyle{r}=frac{18}{{{6}-{6} cos{theta}}} and graph it by the use of graphing utility.

Question
Conic sections
To calculate: The sections represented by the polar equation $$\displaystyle{r}=\frac{18}{{{6}-{6} \cos{\theta}}}$$ and graph it by the use of graphing utility.

2021-03-08
Step 1
The equation $$\displaystyle{r}=\frac{18}{{{6}-{6} \cos{\theta}}}$$ is not in standard form as the constant tern in the denominator is not 1.
To obtain 1 as the constant term in the denominator divide the numerator and denominator by 4.
Thus,
$$\displaystyle{r}=\frac{18}{{{6}-{6} \cos{\theta}}}=\frac{3}{{{1}- \cos{\theta}}}$$
The equation $$\displaystyle{r}=\frac{3}{{{1}- \cos{\theta}}}$$
is in the standard form of $$\displaystyle{r}=\frac{{{e}{p}}}{{{1}-{e} \cos{\theta}}}.$$
On comparing the provided equation to the standard form it can be obtained that $$\displaystyle{e}={1}.$$
Since $$\displaystyle{e}={1},$$ thus the given polar equation represents a parabola.
Step 2
Now, use Ti-83 to plot the graph of the function:
a) Press the [MODE] key then select the polar function and the radian mode.
b) Press the [Y=] key, then there will appear the equations for y.
c) Enter the equations in $$\displaystyle{r}_{{1}}$$.
Here, $$\displaystyle{r}_{{1}}=\frac{18}{{{6}-{6} \cos{\theta}}}.$$
d) Press [WINDOW] and then edit the values as:
$$\displaystyle{X}\min=-{7},{X}\max={7},{X}\text{Scale}={1},{Y}\min=-{7}{\quad\text{and}\quad}{Y}\text{Scale}={1}$$
e) press the [GRAPH] key to plot the graph.
The graph:

### Relevant Questions

Polar equations for conic sections Graph the following conic sections, labeling vertices, foci, directrices, and asymptotes (if they exist). Give the eccentricity of the curve. Use a graphing utility to check your work. $$\displaystyle{r}=\ {\frac{{{10}}}{{{5}\ +\ {2}\ {\cos{\theta}}}}}$$
Identify the conic section given by the polar equation $$\displaystyle{r}=\frac{4}{{{1}- \cos{\theta}}}$$ and also determine its directrix.
Solve, a. If necessary, write the equation in one of the standard forms for a conic in polar coordinates $$r = \frac{6}{2 + sin \theta}$$ b. Determine values for e and p. Use the value of e to identify the conic section. c. Graph the given polar equation.
Solve for the equation in standard form of the following conic sections and graph the curve on a Cartesian plane indicating important points.
1. An ellipse passing through (4, 3) and (6, 2)
2. A parabola with axis parallel to the x-axis and passing through (5, 4), (11, -2) and (21, -4)
3. The hyperbola given by $$\displaystyle{5}{x}^{2}-{4}{y}^{2}={20}{x}+{24}{y}+{36}.$$
Solve, a.Determine the conic section of the polar equation $$\displaystyle{r}=\frac{8}{{{2}+{2} \sin{\theta}}}$$ represents. b. Describe the location of a directrix from the focus located at the pole.
The eccentricities of conic sections with one focus at the origin and the directrix corresponding and sketch a graph:
$$\displaystyle{e}={2},$$
$$directrix \displaystyle{r}=-{2} \sec{\theta}.$$
The polar equation of the conic with the given eccentricity and directrix and focus at origin: $$\displaystyle{r}={41}\ +\ {\cos{\theta}}$$
The reason ehy the point $$\displaystyle{d}{i}{s}{p}{l}{a}{y}{s}{t}{y}\le{\left\lbrace{\left(-{\left\lbrace{1}\right\rbrace},{\frac{{{\left\lbrace{\left\lbrace{3}\right\rbrace}\pi\right\rbrace}}}{{{\left\lbrace{2}\right\rbrace}}}}\right)}\right\rbrace}$$ lies on the polar graph $$\displaystyle{d}{i}{s}{p}{l}{a}{y}{s}{t}{y}\le{\left\lbrace{r}\right\rbrace}={\left\lbrace{1}\right\rbrace}+{\cos{{\left\lbrace\theta\right\rbrace}}}$$ even though it does not satisfy the equation.
$$\displaystyle{2}\sqrt{{3}}{x}^{2}-{6}{x}{y}+\sqrt{{3}}{x}+{3}{y}={0}$$
b) Use a rotation of axes to eliminate the xy-term. (Write an equation in XY-coordinates. Use a rotation angle that satisfies $$\displaystyle{0}\le\varphi\le\pi\text{/}{2}$$)