Question

Derive the equation of the parabola with a focus at (2, 4) and a directrix of y = 8.

Answer 1

bearing in mind that the vertex of the parabola is half-way between the directrix and the focus point

what do you think are the coordinates of the vertex?

Answer 2

The vertex would be (2, 0)?

Answer 3

not quite check.... see what would be the vertex coordinates

Answer 4

(2,4)

Answer 5

"ith a \(\bf \text{focus at (2, 4)}\) and a directrix of y = 8."

Answer 6

that's just the given focus point
but the vertex is half-way between the focus point and the directrix

Answer 7

Oh! So 8-4 = 4/2 is 2 in each direction. The vertex would be (2,6) with a negative slope since it is opening down?

Answer 8

yeap so
now we know the vertex is at (2,6) and the parabola is indeed opening downwards thus

noticd the distnace from the vertex to the focus or the directrix, since it is the same

2 units up and 2 units down

thus \(\bf (x-{\color{brown}{ h}})^2=4{\color{blue}{ p}}(y-{\color{brown}{ k}})
\\ \quad \\
vertex \ (h,k)\qquad p=\textit{distance from the vertex to focus or directrix}
\\ \quad \\
(x-{\color{brown}{ 2}})^2=4({\color{blue}{ 2}})(y-{\color{brown}{ 6}})\)