Question

Graph the cllipse \({\rm{r = 2/(4 - 3cos\theta )}}\) and its dircctrix. Also graph the ellipse obtained by rotation about the origin through an angle \({\rm{2\pi /3}}\).

Short answer

The graph is shown below:

Step-by-step solution

Definition of Concept

Polar curve: A polar curve is a shape made with the polar coordinate system. Polar curves are defined by points that vary in distance from the origin (the pole) based on the angle measured off the positive x-axis.

Graph the cllipse and its dircctrix

Using the following theorem:

A polar equation of the form \({\rm{r = }}\frac{{{\rm{ed}}}}{{{\rm{1 \pm esin\theta }}}}\) or \({\rm{r = }}\frac{{{\rm{ed}}}}{{{{\rm{1}}_{{\rm{ \pm ecos\theta }}}}}}\) represents a conic section with eccentricity e. The conic is an ellipse if\({\rm{e < 1}}\) , a parabola if\({\rm{e = 1}}\), or a hyperbola if \({\rm{e > 1}}\).

Considering the given information:

The polar equation is\({\rm{r = }}\frac{{\rm{2}}}{{{\rm{4 - 3cos\theta }}}}\).

The eccentricity of the given equation should be calculated.

The polar equation for the given equation is either\({\rm{r = }}\frac{{{\rm{ed}}}}{{{{\rm{1}}_{{\rm{ \pm e}}}}{\rm{sin\theta }}}}\)or\({\rm{r = }}\frac{{{\rm{ed}}}}{{{\rm{1 \pm ecos\theta }}}}\).

Because the given polar equation is\({\rm{r = }}\frac{{\rm{2}}}{{{\rm{4 - 3cos\theta }}}}\), divide it by 4 as follows.

\(\begin{aligned}{l}{\rm{r = }}\frac{{\rm{2}}}{{{\rm{4 - 3cos\theta }}}}\\{\rm{r = }}\frac{{\frac{{\rm{2}}}{{\rm{4}}}}}{{\frac{{\rm{4}}}{{\rm{4}}}{\rm{ - }}\frac{{\rm{3}}}{{\rm{4}}}{\rm{cos\theta }}}}\\{\rm{r = }}\frac{{\frac{{\rm{1}}}{{\rm{2}}}}}{{{\rm{1 - }}\frac{{\rm{3}}}{{\rm{4}}}{\rm{cos\theta }}}}\end{aligned}\)

As a result, the eccentricity of the given equation is\(\frac{{\rm{3}}}{{\rm{4}}}\).

Compare this equation to the standard equation\({\rm{r = }}\frac{{{\rm{ed}}}}{{{\rm{1 \pm ecos\theta }}}}\).

Equate the numerator as,

\(\begin{aligned}{l}{\rm{ed = }}\frac{{\rm{1}}}{{\rm{2}}}\\\frac{{\rm{3}}}{{\rm{4}}}{\rm{ \times d = }}\frac{{\rm{1}}}{{\rm{2}}}\\{\rm{d = }}\frac{{\rm{1}}}{{\rm{2}}}{\rm{ \times }}\frac{{\rm{4}}}{{\rm{3}}}\\{\rm{d = }}\frac{{\rm{2}}}{{\rm{3}}}\end{aligned}\)

Therefore, the equation of directrix is\({\rm{x = - }}\frac{{\rm{2}}}{{\rm{3}}}\).

Sketch the graph of the conic \({\rm{r = }}\frac{{\rm{2}}}{{{\rm{4 - 3cos\theta }}}}\) and its directrix using an online graphing calculator, as shown in Figure 1.

From Figure 1, it is observed that the given figure shows the ellipse with its directrix\({\rm{x = - }}\frac{{\rm{2}}}{{\rm{3}}}\).

Next, the conic obtained by rotating the curve\({\rm{r = }}\frac{{\rm{2}}}{{{\rm{4 - 3cos\theta }}}}\)about the origin through the angle\(\frac{{{\rm{2\pi }}}}{{\rm{3}}}\).

Write the polar equation for the conic which is rotated with angle\(\frac{{{\rm{2\pi }}}}{{\rm{3}}}\).

\({\rm{r = }}\frac{{\rm{2}}}{{{\rm{4 - 3cos}}\left( {{\rm{\theta - }}\frac{{{\rm{2\pi }}}}{{\rm{3}}}} \right)}}\)

Use online graphing calculator and sketch the graph of the conic \({\rm{r = }}\frac{{\rm{2}}}{{{\rm{4 - 3cos\theta }}}}\) which is rotated with about the origin through the angle \(\frac{{{\rm{2\pi }}}}{{\rm{3}}}\)as shown below in Figure 2 .

Therefore, from Figure 2, it is observed that the graph represents a ellipse where the angle is changed in to \({\rm{\theta - }}\frac{{{\rm{2\pi }}}}{{\rm{3}}}{\rm{. }}\)