# 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.

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: