1. **Problem Statement:** Write the equation of the parabola in conic form that opens to the right, has a focus of (2, -3), and a focal chord length of 32. 2. **Recall the formula for a parabola opening right:** $$4p(y-k)^2 = (x-h)^2$$ where $(h,k)$ is the vertex, and $p$ is the distance from the vertex to the focus. 3. **Given:** - Focus: $(2, -3)$ - Focal chord length: 32 4. **Find $p$:** The focal chord length is $4p$, so $$4p = 32 \implies p = 8$$ 5. **Find the vertex $(h,k)$:** Since the parabola opens right, the vertex lies midway between the focus and the directrix. The directrix is vertical and located at $x = h - p$. Given the focus at $(2, -3)$ and $p=8$, the vertex is at $$h = 2 - 8 = -6, \quad k = -3$$ 6. **Write the equation:** $$4p(y - k)^2 = (x - h)^2 \implies 4 \times 8 (y + 3)^2 = (x + 6)^2$$ Simplify: $$32(y + 3)^2 = (x + 6)^2$$ --- 7. **Problem Statement:** Determine the equation of the parabola in conic form with a focus of $(2, 7)$ and a directrix of $x = -18$. 8. **Recall the formula for a parabola opening right or left:** $$4p(y-k)^2 = (x-h)^2$$ 9. **Find the vertex $(h,k)$:** The vertex lies midway between the focus and directrix. $$h = \frac{2 + (-18)}{2} = -8, \quad k = 7$$ 10. **Find $p$:** Distance from vertex to focus: $$p = 2 - (-8) = 10$$ 11. **Write the equation:** $$4p(y - k)^2 = (x - h)^2 \implies 4 \times 10 (y - 7)^2 = (x + 8)^2$$ Simplify: $$40(y - 7)^2 = (x + 8)^2$$ **Final answers:** - For problem 9: $$32(y + 3)^2 = (x + 6)^2$$ - For problem 10: $$40(y - 7)^2 = (x + 8)^2$$