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

Sep 20, 2017

Equation of parabola is $y = - \frac{1}{20} {x}^{2}$

#### 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 $y = a {\left(x - 0\right)}^{2} + 0 \mathmr{and} y = a {x}^{2}$

Focus is at $\left(0 , - 5\right)$ , 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 $d = \frac{1}{4 | a |} \therefore | a | = \frac{1}{4 \cdot 5} = \frac{1}{20} \therefore a = - \frac{1}{20}$

Hence equation of parabola is $y = - \frac{1}{20} {x}^{2}$

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