Question

What is the vertex form of the equation of the parabola with a focus at (20,-10) and a directrix of `y=15`?

Answer 1

Equation of parabola is `y = -1/50(x-20)^2+2.5`.

Explanation:

Focus is at `(20 ,-10)` and directrix is `y=15`

Vertex is equidistant from focus and directrix and in midway

between them. So vertex is at `(20, (15-10)/2=2.5) or (20, 2.5)`

Equation of Parabola in vertex form is `y= a(x-h)^2+k` or

`y = a(x-20)^2+2.5` . Focus is below the vertex so parabola

opens downwards `:.a` is negative . Distance of directrix from

vertex is `d = 1/(4|a|) ;d = (15-2.5)=12.5 :. |a| = 1/(4d)`

`= 1/(4*12.5)=1/50 :. a = - 1/50`.

Hence equation of parabola is `y = -1/50(x-20)^2+2.5`

graph{-1/50(x-20)^2+2.5 [-160, 160, -80, 80]}