To find the equation of a parabola with its vertex at the origin (0,0) and a directrix at y = 15, we start by recalling the definition of a parabola: it is the set of all points that are equidistant from a point called the focus and a line called the directrix.
1. Identifying the focus: Given that the directrix is y = 15, the focus must be below the directrix in order for the parabola to open downward. The distance from the vertex to the directrix is 15 units. Since the vertex is at the origin, the focus will be located at (0, -p), where p is the distance from the vertex to the focus. Because the distance to the directrix is 15, this means that:
p = -15, placing the focus at the coordinates (0, -15).
2. Finding the equation: The standard form of a parabola that opens downwards, with the vertex at the origin, is given by:
y = - rac{1}{4p} x^2
Substituting p = -15 into the equation, we get:
y = - rac{1}{4(-15)} x^2
This simplifies to:
y = rac{1}{60} x^2
3. Final equation: Therefore, the equation of the parabola with its vertex at the origin and a directrix of y = 15 is:
y = - rac{1}{60} x^2
This equation describes a parabola that opens downwards, making it consistent with the position of the directrix and the focus you have established.