Parabolas Ii Aac8D8

1. **State the problem:** We need to find the focus and directrix of the parabola given by the equation $$(x - 5)^2 = 8(y - 1)$$. 2. **Identify the form:** This equation is in the form $$(x - h)^2 = 4p(y - k)$$ where $(h, k)$ is the vertex. 3. **Find the vertex:** From the equation, $h = 5$ and $k = 1$, so the vertex is at $$(5, 1)$$. 4. **Find $p$:** Compare $4p$ with $8$ to get $4p = 8$, so $$p = \frac{8}{4} = 2$$. 5. **Interpret $p$:** Since the parabola opens upwards (because the $x$ term is squared and $p > 0$), the focus is $$p$$ units above the vertex, and the directrix is $$p$$ units below the vertex. 6. **Calculate the focus:** The focus coordinates are $$(h, k + p) = (5, 1 + 2) = (5, 3)$$. 7. **Calculate the directrix:** The directrix is the line $$y = k - p = 1 - 2 = -1$$. **Final answer:** Focus: $$(5, 3)$$ Directrix: $$y = -1$$