Description:

Consider the polar coordinates as \(\displaystyle{P}{\left({r}.\theta\right)}\).

Here, r is the directed distance from origin Oto point Pand \(\displaystyle\theta\) is the directed angle from initial ray to OP.

From the given equation, the range of r is from —2 to \(\displaystyle—\propto\) and 0s fixed at \(\displaystyle{\frac{{{2}\pi}}{{{3}}}}\). Therefore, the polar point traces a straight line at an angle of \(\displaystyle{\frac{{{2}\pi}}{{{3}}}}\) from the initial ray (x-axis) and the line also satisfies the range of r.

From the analysis, draw the sets of points whose polar coordinates satisfy the given polar equation as shown in Figure 1.

\(\displaystyle\theta={\frac{{{2}\pi}}{{{3}}}},\)

\(\displaystyle{r}\le-{2}\)

Consider the polar coordinates as \(\displaystyle{P}{\left({r}.\theta\right)}\).

Here, r is the directed distance from origin Oto point Pand \(\displaystyle\theta\) is the directed angle from initial ray to OP.

From the given equation, the range of r is from —2 to \(\displaystyle—\propto\) and 0s fixed at \(\displaystyle{\frac{{{2}\pi}}{{{3}}}}\). Therefore, the polar point traces a straight line at an angle of \(\displaystyle{\frac{{{2}\pi}}{{{3}}}}\) from the initial ray (x-axis) and the line also satisfies the range of r.

From the analysis, draw the sets of points whose polar coordinates satisfy the given polar equation as shown in Figure 1.

\(\displaystyle\theta={\frac{{{2}\pi}}{{{3}}}},\)

\(\displaystyle{r}\le-{2}\)