Question

Write the equation of the parabola with a focus (1,2) and directrix y = 1.

Answer 1

Im not sure how to start.

Answer 2

1?

Answer 3

okay the formula you put up above?

Answer 4

oh okay, onne second and then ill tell you what I got:)

Answer 5

one may note that 1,2 is the focus point, not the vertex point

Answer 6

oh wait what does that mean then?

Answer 7

well... she maybe crazy, but still beautiful, so tis ok =)

Answer 8

haha.

Answer 9

im seriously confused though...

Answer 10

Im not sure... Im sorry, this is new to me.

Answer 11

hmmm well.. have you covered parabolas yet?

Answer 12

Yes, briefly.

Answer 13

well.. it turns out that
the focus point is "p" distance from the vertex
and the directrix is "p" distance also from the vertex

so.. the vertex is really half-way between both, since the focus and directrix are the same distance from the vertex

keeping in mind that, the parabola opens up towards the focus point

Answer 14

Okay, well crazyandbeautiful helped me out and said that (1,3/2) was the vertex. haha.

Answer 15

hmmm how did she get 1, 3/2 though?

Answer 16

haha no you dont need to its okay! @Crazyandbeautiful

Answer 17

okay so its always halfway between the focus point and directrix?

Answer 18

yeap, they're always "p" distance from the vertex

Answer 19

Okay, i get that now:)

Answer 20

Where do I go from there?

Answer 21

so... get the vertex, get "p" and plug and chug \(\large{ \begin{array}{llll}
(y-{\color{blue}{ k}})^2=4{\color{purple}{ p}}(x-{\color{brown}{ h}}) \\
(x-{\color{brown}{ h}})^2=4{\color{purple}{ p}}(y-{\color{blue}{ k}})\\
\end{array}

\qquad
\begin{array}{llll}
vertex\ ({\color{brown}{ h}},{\color{blue}{ k}})\\
{\color{purple}{ p}}=\textit{distance from vertex to }\\
\qquad \textit{ focus or directrix}
\end{array}}\)