focus: (0,p)
vertex: if not given, then it is assumed (0,0)
Directrix: k+p or k-p, i'm not sure
I think these are the general equations for the focus, vertex, and directrix of a parabola, but i'm not sure. could someone please check this? thanks.
Yes, you are correct about the general equations for the focus, vertex, and directrix of a parabola. Let's go through each one to clarify:
1. Focus: The focus of a parabola with the vertex at (0, 0) is located at (0, p). The focus represents the fixed point inside the parabola where all the reflected rays converge or appear to diverge from.
2. Vertex: If the vertex of a parabola is not given explicitly, it can be assumed to be (0, 0). The vertex is the point where the parabola reaches its minimum or maximum value and represents the lowest or highest point on the curve.
3. Directrix: The directrix of a parabola can be either k + p or k - p, depending on whether the parabola opens upwards or downwards. It is a straight line perpendicular to the axis of symmetry that is located on the opposite side of the parabola from the focus. The distance between the directrix and the vertex is equal to the distance between the focus and the vertex.
So, to summarize, if the vertex is at (0, 0), the focus will be at (0, p), and the directrix can be either k + p or k - p depending on the orientation of the parabola.