1. **State the problem:** Write an equation of a parabola with a vertex at the origin and a directrix at $y=5$. 2. **Recall the definition and formula:** A parabola is the set of points equidistant from the focus and the directrix. 3. Since the vertex is at the origin $(0,0)$ and the directrix is horizontal at $y=5$, the parabola opens downward because the vertex is below the directrix. 4. The vertex form of a parabola with vertical axis is $$ (x - h)^2 = 4p(y - k) $$ where $(h,k)$ is the vertex and $p$ is the distance from the vertex to the focus (positive if opening upward, negative if downward). 5. Here, $h=0$, $k=0$, and the directrix is $y=5$, so the distance from vertex to directrix is $|p|=5$. 6. Since the parabola opens downward, $p = -5$. 7. Substitute into the formula: $$ x^2 = 4(-5)(y - 0) $$ $$ x^2 = -20y $$ 8. This is the equation of the parabola with vertex at origin and directrix $y=5$. **Final answer:** $$\boxed{x^2 = -20y}$$