To solve for the equation of the directrix and the vertex of the given polar equation [tex]\( r = \frac{8}{2 - 2 \sin \theta} \)[/tex], we need to recognize that the equation describes a conic section in polar coordinates. Conic sections in polar coordinates typically take the form:
[tex]\[ r = \frac{l}{1 - e \sin \theta} \text{ or } r = \frac{l}{1 - e \cos \theta} \][/tex]
where [tex]\( e \)[/tex] is the eccentricity and [tex]\( l \)[/tex] is a parameter related to the conic section.
We can first simplify the given equation:
[tex]\[ r = \frac{8}{2(1 - \sin \theta)} = \frac{4}{1 - \sin \theta} \][/tex]
Thus, we identify:
[tex]\[ r = \frac{4}{1 - \sin \theta} \][/tex]
Comparing this with the standard form [tex]\( \frac{l}{1 - e \sin \theta} \)[/tex], we find:
- [tex]\( l = 4 \)[/tex]
- [tex]\( e = 1 \)[/tex] (since the coefficient of [tex]\(\sin \theta\)[/tex] is 1)
### Directrix:
For a conic section given by [tex]\( r = \frac{l}{1 - e \sin \theta} \)[/tex], the directrix can be found using the relationship that, for a vertical directrix (parallel to the [tex]\( y \)[/tex]-axis):
[tex]\[ y = \frac{l}{e} \][/tex]
Given that:
- [tex]\( l = 4 \)[/tex]
- [tex]\( e = 1 \)[/tex]
The equation of the directrix is:
[tex]\[ y = \frac{4}{1} = 4 \][/tex]
### Vertices:
The vertex of the conic section in polar coordinates occurs when the denominator of the fraction achieves its minimum value. This happens when [tex]\( \theta = \frac{\pi}{2} \)[/tex] or [tex]\( \theta = \frac{3\pi}{2} \)[/tex] because [tex]\(\sin \theta\)[/tex] reaches its maximum magnitude of [tex]\(\pm 1\)[/tex] at these angles.
1. For [tex]\( \theta = \frac{\pi}{2} \)[/tex]:
[tex]\[ r = \frac{4}{1 - \sin \frac{\pi}{2}} = \frac{4}{1 - 1} = \text{undefined} \][/tex]
This indicates an asymptote rather than a vertex because [tex]\( r \)[/tex] approaches infinity.
2. For [tex]\( \theta = \frac{3\pi}{2} \)[/tex]:
[tex]\[ r = \frac{4}{1 - \sin \frac{3\pi}{2}} = \frac{4}{1 - (-1)} = \frac{4}{2} = 2 \][/tex]
Thus, the vertex on the conic section, in polar coordinates, is located at [tex]\( (\theta, r) = \left( \frac{3\pi}{2}, 2 \right) \)[/tex].
### Summary:
- The directrix of the conic section is the line [tex]\( y = 4 \)[/tex].
- The vertex of the conic section is at [tex]\( r = 2 \)[/tex] when [tex]\( \theta = \frac{3\pi}{2} \)[/tex] or in other words, at [tex]\( ( \frac{3\pi}{2}, 2 ) \)[/tex] in polar coordinates.
