Question

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. \(\displaystyle{r}=\ {\frac{{{10}}}{{{5}\ +\ {2}\ {\cos{\theta}}}}}\)

Answer 1

Step 1 GIven polar equation is: \(\displaystyle{r}=\ {\frac{{{10}}}{{{5}\ +\ {2}\ {\cos{\theta}}}}}\) The graph of the given polar equation is: Step 2 Therefore the given polar equation represents an ellipse with major axis as the x-axis, center at \(\displaystyle{\left(-{1},\ {0}\right)}.\)
\(\displaystyle{r}=\ {\frac{{{10}}}{{{\left({5}\ +\ {2}\ {\cos{\theta}}\right\rbrace}}}}\)
\(\displaystyle{r}=\ {\frac{{{2}}}{{{\left[{1}\ +\ {\left({\frac{{{2}}}{{{5}}}}\right)}\ {\cos{\theta}}\right]}}}}\)
Eccentricity \(\displaystyle{e}=\ {\frac{{{2}}}{{{5}}}}\ {<}\ {1}\)</span> Therefore the given equation represents an ellipse. Now we get, \(\displaystyle{2}={e}{p}\)
\(\displaystyle{2}=\ {\left({\frac{{{2}}}{{{5}}}}\right)}{p}\)
\(\displaystyle{p}={5}\) The directrix is \(\displaystyle{y}={5}.\)