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
asked 2021-03-07
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.

Answers (1)

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:
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