Question

What is the equation of the parabola with a focus at (7,3) and a directrix of y= 4?

Answer 1

The equation of parabola is `y= - 0.5 (x-7)^2+3.5`

Explanation:

Focus is at `(7,3)` , Directrix is `y =4` . The vertex is at midway

between focus and directrix .

So vertex is at `(7,(3+4)/2) or (7,3.5)` , The distance between

vertex and directrix is `d= 4-3.5 = 0.5` . Here directrix is above

vertex , so parabola opens downward and `a` is negative.

`d =1/(4|a|) or 0.5 = 1/(4|a|) :. |a| = 1/2 , a = - 0.5`

The equation of parabola is `y= a(x-h)^2+k ; (h,k)`

being vertex. Here `h= 7 , k= 3.5 ,a = - 0.5` . Hence

the equation of parabola is `y= - 0.5 (x-7)^2+3.5`

graph{-0.5(x-7)^2 +3.5 [-20, 20, -10, 10]}