Chapter 10, Problem 55RE

To determine

Tofind:the conic of the polar equation and graph the polar equation.

Answer to Problem 55RE

The conic is hyperbolawith transverse axis perpendicular to directrix.

Explanation of Solution

Given:

  $\text{r=}\frac{6}{\text{2-sinθ}}$ .

Concept used:

For a conic with eccentricity e

If $\text{0}\le \text{e}\le \text{1}$ the conic is ellipse.

If $e=1$ then the conic is parabola.

If $e>1$ ,then the conic is hyperbola.

For a conic with focus at origin and Directrix is $\text{x=±p}$ .

Then Polar equation of the conic is

  $\text{r=}\frac{\text{ep}}{\text{1±esinθ}}$ .

For a conic with focus at origin and Directrix is $\text{y=±p}$ .

Then Polar equation of the conic is

  $\text{r=}\frac{\text{ep}}{\text{1±ecosθ}}$ .

Calculation:

  $\text{r=}\frac{6}{\text{2-sinθ}}$ .

  $\text{r=}\frac{\text{3}}{\text{1-}\frac{\text{1}}{\text{2}}\text{sinθ}}$ .

Comparing this with polar equation:

  $\text{r=}\frac{\text{ep}}{\text{1±esinθ}}$

  $\text{ep=3}$ .

  $\left(\text{}\frac{\text{1}}{\text{2}}\right)\text{p=3}$ .

  $p=6$ .

Here $\text{e=}\frac{1}{2}\text{ and p=6}$ .

If $\text{0}\le \text{e}\le \text{1}$ the conic is ellipse.

So, the conic isellipse with the major axis perpendicular to directrix.

The directrix is parallel to polar axis at a distance 6 units below the pole.

The graph of the polar equation $\text{r=}\frac{6}{\text{2-sinθ}}$ :

  

Hence, the conic is ellipse with the major axis perpendicular to directrix.