Question

practice using the focus and directrix to define a parabola. when the focus and directrix are used to derive the equation of a parabola, two distances were set equal to each other. $sqrt{(x - x)^{2}+(y - (-p))^{2}}=sqrt{(x - 0)^{2}+(y - p)^{2}}$. the distance between the directrix and the focus is set equal to the distance between the and the same point on the parabola. x directrix focus origin x - axis

Explanation:

Step1: Recall parabola definition

A parabola is defined as the set of all points that are equidistant from a fixed - point (the focus) and a fixed - line (the directrix).

Step2: Identify the equal distances

The distance between the focus $F(0,p)$ and a point $P(x,y)$ on the parabola is set equal to the distance between the directrix $y = - p$ and the point $P(x,y)$ on the parabola.

Answer:

The distance between the focus and the point on the parabola is set equal to the distance between the directrix and the same point on the parabola. So the first blank is "the focus", and the second blank is "directrix".