Question: Find an equation of the parabola whose vertex is at the origin and the focus at $(0, 8)$.

Find an equation of the parabola whose vertex is at the origin and the focus at $(0, 8)$.

Solution

To find the equation of the parabola with vertex at the origin and focus at \((0, 8)\): Determine the value of \( p \), the distance from the vertex to the focus. \[ p = 8 \] Use the standard form of a vertical parabola with vertex at the origin: \[ y = \frac{1}{4p}x^2 \] Substitute \( p = 8 \) into the equation: \[ y = \frac{1}{32}x^2 \]