1. **State the problem:** We want to find the coordinates of points on the conic given by the polar equation $r = \frac{4}{4 - 2 \cos \theta}$ to help sketch the graph. 2. **Recall the polar to Cartesian conversion formulas:** $$x = r \cos \theta$$ $$y = r \sin \theta$$ 3. **Calculate $r$ for specific values of $\theta$:** - At $\theta = 0$: $$r = \frac{4}{4 - 2 \cos 0} = \frac{4}{4 - 2 \times 1} = \frac{4}{2} = 2$$ - At $\theta = \frac{\pi}{2}$: $$r = \frac{4}{4 - 2 \cos \frac{\pi}{2}} = \frac{4}{4 - 2 \times 0} = \frac{4}{4} = 1$$ - At $\theta = \pi$: $$r = \frac{4}{4 - 2 \cos \pi} = \frac{4}{4 - 2 \times (-1)} = \frac{4}{6} = \frac{2}{3}$$ - At $\theta = \frac{3\pi}{2}$: $$r = \frac{4}{4 - 2 \cos \frac{3\pi}{2}} = \frac{4}{4 - 2 \times 0} = 1$$ 4. **Convert these polar coordinates to Cartesian coordinates:** - For $\theta = 0$: $$x = 2 \times \cos 0 = 2, \quad y = 2 \times \sin 0 = 0$$ - For $\theta = \frac{\pi}{2}$: $$x = 1 \times \cos \frac{\pi}{2} = 0, \quad y = 1 \times \sin \frac{\pi}{2} = 1$$ - For $\theta = \pi$: $$x = \frac{2}{3} \times \cos \pi = -\frac{2}{3}, \quad y = \frac{2}{3} \times \sin \pi = 0$$ - For $\theta = \frac{3\pi}{2}$: $$x = 1 \times \cos \frac{3\pi}{2} = 0, \quad y = 1 \times \sin \frac{3\pi}{2} = -1$$ 5. **Summary:** These points $(2,0)$, $(0,1)$, $(-\frac{2}{3},0)$, and $(0,-1)$ help us understand the shape and size of the conic before sketching. **Final answer:** The coordinates at key angles are $(2,0)$, $(0,1)$, $(-\frac{2}{3},0)$, and $(0,-1)$, calculated by converting from polar to Cartesian using $x = r \cos \theta$ and $y = r \sin \theta$.