When the focus and directrix are used to derive the equation of a parabola, two distances are set equal to each other.
To understand this, let's break it down step-by-step:
1. Identify the Focus and the Directrix:
- The focus of the parabola is a point [tex]\((0, p)\)[/tex].
- The directrix is a horizontal line [tex]\(y = -p\)[/tex].
2. Consider a Point on the Parabola:
- Let [tex]\((x, y)\)[/tex] be a generic point on the parabola. By definition, the distance from this point to the focus and the vertical distance to the directrix must be equal.
3. Distance from the Point to the Focus:
- The distance between the point [tex]\((x, y)\)[/tex] and the focus [tex]\((0, p)\)[/tex] is given by:
[tex]\[
\sqrt{(x - 0)^2 + (y - p)^2}
\][/tex]
4. Distance from the Point to the Directrix:
- The distance between the point [tex]\((x, y)\)[/tex] and the directrix [tex]\(y = -p\)[/tex] is the vertical distance given by [tex]\(y - (-p)\)[/tex], which simplifies to:
[tex]\[
|y + p|
\][/tex]
- Since [tex]\(y\)[/tex] can be both above and below [tex]\(-p\)[/tex], we take the absolute value to ensure the distance is positive.
5. Set the Two Distances Equal:
- According to the definition of a parabola, these two distances must be equal:
[tex]\[
\sqrt{(x - 0)^2 + (y - p)^2} = |y + p|
\][/tex]
In the context of your original equations where the distances are:
[tex]\[
\sqrt{(x - x)^2 + (y - (-p))^2} = \sqrt{(x - 0)^2 + (y - p)^2}
\][/tex]
These equation pieces correspond to:
1. Distance Between the Directrix and the Point on the Parabola:
[tex]\[ \sqrt{(x - x)^2 + (y - (-p))^2} \][/tex]
2. Distance Between the Focus and the Same Point on the Parabola:
[tex]\[ \sqrt{(x - 0)^2 + (y - p)^2} \][/tex]
Therefore, we can fill in the blanks in the statement as follows:
The distance between the directrix and the point on the parabola is set equal to the distance between the focus and the same point on the parabola.
