1. **State the problem:** We need to sketch the graph of the conic given by the polar equation $$r = \frac{4}{4 - 2 \cos \theta}$$ and clearly label the directrices. 2. **Recall the form of the conic in polar coordinates:** The general form for a conic with focus at the pole is $$r = \frac{ed}{1 + e \cos \theta}$$ or $$r = \frac{ed}{1 - e \cos \theta}$$ where $e$ is the eccentricity and $d$ is the distance from the focus to the directrix. 3. **Rewrite the given equation:** $$r = \frac{4}{4 - 2 \cos \theta} = \frac{4}{4(1 - \frac{1}{2} \cos \theta)} = \frac{1}{1 - \frac{1}{2} \cos \theta}$$ This matches the form $$r = \frac{ed}{1 - e \cos \theta}$$ with $$ed = 1$$ and $$e = \frac{1}{2}$$. 4. **Identify parameters:** - Eccentricity: $$e = \frac{1}{2}$$ - Since $$ed = 1$$, then $$d = \frac{1}{e} = 2$$. 5. **Directrix location:** - For $$r = \frac{ed}{1 - e \cos \theta}$$, the directrix is the vertical line $$x = d = 2$$. 6. **Sketching the graph:** - The conic is an ellipse (since $$e < 1$$). - The focus is at the pole (origin). - The directrix is the vertical line $$x = 2$$. - The curve is closer to the focus when $$\theta = 0$$ (since $$\cos 0 = 1$$), and farther when $$\theta = \pi$$. 7. **Plot key points:** - At $$\theta = 0$$, $$r = \frac{4}{4 - 2(1)} = \frac{4}{2} = 2$$. - At $$\theta = \pi$$, $$r = \frac{4}{4 - 2(-1)} = \frac{4}{6} = \frac{2}{3}$$. 8. **Label the directrix:** Draw the vertical line $$x = 2$$ on the polar plot. **Final answer:** The sketch is an ellipse with focus at the origin, directrix at $$x=2$$, and eccentricity $$\frac{1}{2}$$. The curve is closer to the focus at $$\theta=0$$ and farther at $$\theta=\pi$$.