Question

Sketchthe polar curve.

\({\rm{r = }}\frac{{\rm{3}}}{{{\rm{2 - 2cos\theta }}}}\)

Short answer

The polar curve is:

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.

Sketch the polar curve

Considering the given information:

Polar equation for the variable

\({\rm{r = }}\frac{{\rm{3}}}{{{\rm{2 - 2cos\theta }}}}\)

The given polar equation is of the form,

\({\rm{r = }}\frac{{{\rm{ed}}}}{{{\rm{1 \pm ecos\theta }}}}{\rm{ (1) }}\)

Note that\({\rm{r = }}\frac{{\rm{3}}}{{{\rm{2 - 2cos\theta }}}}\)becomes,

\({\rm{r = }}\frac{{\frac{{\rm{3}}}{{\rm{2}}}}}{{{\rm{1 - cos\theta }}}}\)

Comparing the equation (1) with\({\rm{r = }}\frac{{\rm{3}}}{{{\rm{2 - 2cos\theta }}}}\)

Note that\({\rm{ed = }}\frac{{\rm{3}}}{{\rm{2}}}\)and e=1, therefore\({\rm{d = }}\frac{{\rm{3}}}{{\rm{2}}}\).

Since the conic is of the form\({\rm{r = }}\frac{{{\rm{ed}}}}{{{\rm{1 - ecos\theta }}}}\), the directrix will be of the form x=-d.

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

As eccentricity, e is 1 the conic is a parabola.

Putting the value\({\rm{1}}{{\rm{0}}^{\rm{^\circ }}}\)for\({\rm{\theta }}\)and obtain the value of r as,

\(\begin{aligned}{c}{\rm{r = }}\frac{{\rm{3}}}{{\left( {{\rm{2 - }}\left( {{\rm{2 \times cos}}\left( {{\rm{1}}{{\rm{0}}^{\rm{^\circ }}}} \right){\rm{ \times }}\frac{{\rm{x}}}{{{\rm{180}}}}} \right)} \right)}}\\{\rm{ = 98}}{\rm{.73}}\end{aligned}\)

Make a table of values for\(0 \le \theta \le 2\pi \) , plot the points, then connect them with a smooth curve as shown below. The three points are enough to determine the general shape of the parabola which opens away from the directrix.

Therefore, the required polar curve is shown below: