Write an equation for a parabola in which the set of all points in the plane are equidistant from the focus and line. F(0, –5); y = 5
Accepted Solution
A:
well, "the set of all points in the plane equidistant from the focus and a line", is referring to the focus point of the parabola and the directrix line.
bearing in mind that, both fellows are at a distance "p" from the vertex, that puts the vertex right in the middle of them.
notice, the focus point is below the directrix, meaning, is a vertical parabola, and is also opening downwards, like in the picture below.
from -5 to 5 over the y-axis, there are 10 units, so the "p" distance is 5, so that puts the vertex right at the origin.
since the parabola is opening downwards, "p" is negative, thus -5.
[tex]\bf \textit{parabola vertex form with focus point distance}
\\\\
\begin{array}{llll}
4p(x- h)=(y- k)^2
\\\\
\boxed{4p(y- k)=(x- h)^2}
\end{array}
\qquad
\begin{array}{llll}
vertex\ ( h, k)\\\\
p=\textit{distance from vertex to }\\
\qquad \textit{ focus or directrix}
\end{array}\\\\
-------------------------------\\\\
\begin{cases}
h=0\\
k=0\\
p=-5
\end{cases}\implies 4(-5)(y-0)=(x-0)^2
\\\\\\
-20y=x^2\implies y=-\cfrac{1}{20}x^2[/tex]