To find the equation for the directrix of a parabola with a vertex at the origin and a focus at (0, 20), we first need to understand the basic properties of a parabola. A parabola is defined as the set of all points that are equidistant from a fixed point, called the focus, and a fixed line, known as the directrix.
In this case, the focus is positioned at (0, 20), which lies on the y-axis. For a parabola that opens upwards (which is our case since the focus is located above the vertex), the vertex is at the origin (0, 0).
The distance from the vertex to the focus is called ‘p’, which represents the distance from the vertex to the directrix as well. Here, ‘p’ is equal to 20 since the distance from (0, 0) to (0, 20) is 20 units.
Since the parabola opens upwards, the directrix will be a horizontal line located below the vertex. The equation of the directrix can be determined using the equation:
Directrix: y = k – p
Where ‘k’ is the y-coordinate of the vertex and ‘p’ is the distance from the vertex to the focus. In our case:
y = 0 – 20
This simplifies to:
y = -20
Thus, the equation for the directrix of the parabola with a vertex at the origin and a focus at (0, 20) is:
y = -20
This means that all points on the parabola will be equidistant from the focus at (0, 20) and this directrix line at y = -20.