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 is set equal to the distance between the and the same point on the parabola.

Explanation:

Step1: Recall parabola definition

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

Step2: Identify the distances

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

Answer:

The distance between the directrix \(y=-p\) and the point \(P(x,y)\) on the parabola; focus \(F(0,p)\)