Question

How do you write an equation of a parabola with its vertex at the origin and focus at (0,-5)?

Answer 1

Equation of parabola is `y = -1/20x^2`

Explanation:

Equation of parabola in vertex form is `y=a(x-h)^2+k ; (h,k)`

being vertex , here vertex is at origin ,i.e `h=0 ,k=0` .

So equation of parabola is `y=a(x-0)^2+0 or y=ax^2`

Focus is at `(0,-5)` , which is below the vertex . Vertex is at

mid point between focus and directrix ,which is above the vertex.

The parabola opens downward so `a` is negative here.

Directrix is `y=5` and distance of directrix from vertex is `d=5`

We know `d= 1/(4|a|) :. |a|= 1/(4*5)=1/20 :. a = -1/20`

Hence equation of parabola is `y = -1/20x^2`

graph{-1/20x^2 [-40, 40, -20, 20]} [Ans]