Will a parabola intersect its directrix sometimes, always, or never?
Never
A parabola will intersect its directrix only once. To understand why, let's first define what a parabola and a directrix are.
A parabola is a curve defined by the set of points in a plane that are equidistant from a fixed point called the focus and a fixed line called the directrix. The focus lies on the axis of symmetry, which is a line perpendicular to the directrix. The distance between any point on the parabola and the focus is equal to the distance between that point and the directrix.
Now, let's consider what happens when a parabola intersects its directrix. Suppose there is a point on the parabola that lies on or intersects the directrix. Since the distance between any point on the parabola and the focus is equal to the distance between that point and the directrix, this would imply that the point is equidistant from both the focus and the directrix.
However, this contradicts the definition of a parabola. The defining property of a parabola is that the distance between any point on the parabola and the focus is equal to the distance between that point and the directrix. If a point lies on or intersects the directrix, it cannot simultaneously be equidistant from both the focus and the directrix.
Therefore, a parabola will never intersect its directrix. It will only touch the directrix at a single point, and that point is the vertex of the parabola.