To derive the equation of a parabola, we begin by setting two distances equal to each other as given:
[tex]\[
\sqrt{(x - x)^2 + (y - (-p))^2} = \sqrt{(x - 0)^2 + (y - p)^2}
\][/tex]
The important things to note here are the definitions of focus and directrix:
1. Focus: The fixed point [tex]\((x, -p)\)[/tex].
2. Directrix: The fixed line, in this case, represented by [tex]\(y = p\)[/tex].
For a point [tex]\((x, y)\)[/tex] on the parabola, we compare:
1. The distance from this point to the focus.
2. The distance from this point to the directrix.
Now, the given equation showcases the equal distances setup:
1. Distance between the point [tex]\((x, y)\)[/tex] and the focus [tex]\((0, -p)\)[/tex]:
[tex]\[
\sqrt{(x - 0)^2 + (y - (-p))^2} = \sqrt{x^2 + (y + p)^2}
\][/tex]
2. Distance between the point [tex]\((x, y)\)[/tex] and the directrix [tex]\(y = p\)[/tex]:
[tex]\[
\sqrt{(x - x)^2 + (y - p)^2} = \sqrt{(0)^2 + (y - p)^2} = \sqrt{(y - p)^2}
\][/tex]
Equating these distances:
[tex]\[
\sqrt{x^2 + (y + p)^2} = \sqrt{(y - p)^2}
\][/tex]
Given the setup, we conclude that the distance between the point [tex]\((x, y)\)[/tex] on the parabola and the focus [tex]\((0, -p)\)[/tex] is set equal to the distance between the point [tex]\((x, y)\)[/tex] and the directrix [tex]\(y = p\)[/tex]. These relationships define the characteristic property of a parabola.
