SOLUTION: Find the equation of the parabola with vertex at (5, 4) and focus (5, 2).

Algebra ->  Finance -> SOLUTION: Find the equation of the parabola with vertex at (5, 4) and focus (5, 2).      Log On


   

Question 1190795: Find the equation of the parabola with vertex at (5, 4) and focus (5, 2).
Found 2 solutions by josgarithmetic, greenestamps:
Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
Definition for Parabola using the Distance Formula!
The focus is on the other side of the vertex than the directrix.

Starting a graph you can identify directrix as y=6, or (x,6). All values for x.

Distance between (x,y) and (5,2) equals distance between (x,y) and (x,6).
sqrt%28%28x-5%29%5E2%2B%28y-2%29%5E2%29=sqrt%28%28x-x%29%5E2%2B%28y-6%29%5E2%29
Simplify and put into whatever form you need.
.
.
%28x-5%29%5E2=y%5E2-12y%2B36-%28y%5E2-4y%2B4%29
%28x-5%29%5E2=-8y%2B32
highlight%28%28x-5%29%5E2=-8%28y-4%29%29

Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


The exclamation point in the response from the other tutor suggests that finding the equation using the definition of a parabola is "THE" way to solve the problem.

Very much not so. Finding the equation is much simpler than that.

The focus is below the vertex, so the parabola opens downward. In vertex form, with vertex (h,k), the equation is

y-k=%281%2F%284p%29%29%28x-h%5E2%29

where p is the directed distance from the vertex to the focus.

We are given vertex (5,4) and p = -2, so the equation is

ANSWER: y-4=%28-1%2F8%29%28x-5%29%5E2