Question

A parabola can be drawn given a focus of (8,10) and a directrix of y=4. What can be said about the parabola?

Answer 1

The parabola will have a vertical axis of symmetry, with the focus (8,10) above the directrix y=4. The vertex of the parabola will be the point on the axis of symmetry half-way between the focus and directrix, which is (8,7). The distance from the vertex to the focus will be the same as the distance from the vertex to the directrix, which is 3 units. Therefore, the equation of the parabola can be written in vertex form as:

(x-8)^2 = 4p(y-7)

where p=3 is the distance from the vertex to the focus (or directrix). Simplifying this equation, we get:

(x-8)^2 = 12(y-7)

This is the equation of the parabola with focus (8,10) and directrix y=4.