A parabola can be drawn given a focus of (−7,11) and a directrix of y=−1. What can be said about the parabola?
The vertex of the parabola is located halfway between the focus and the directrix, which is at (−7,5). The axis of symmetry of the parabola is a horizontal line passing through the vertex. Since the directrix is below the vertex, the parabola opens upwards. The distance from the vertex to the focus is equal to the distance from the vertex to the directrix. Therefore, the equation of the parabola can be written in vertex form as:
(x + 7)^2 = 4p(y - 5)
where p is the distance from the vertex to the focus (or from the vertex to the directrix), which is 6. So the equation of the parabola is:
(x + 7)^2 = 24(y - 5)