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

Aug 3, 2017

Equation of parabola is $y = - \frac{1}{50} {\left(x - 20\right)}^{2} + 2.5$.

#### Explanation:

Focus is at $\left(20 , - 10\right)$ and directrix is $y = 15$

Vertex is equidistant from focus and directrix and in midway

between them. So vertex is at $\left(20 , \frac{15 - 10}{2} = 2.5\right) \mathmr{and} \left(20 , 2.5\right)$

Equation of Parabola in vertex form is $y = a {\left(x - h\right)}^{2} + k$ or

$y = a {\left(x - 20\right)}^{2} + 2.5$ . Focus is below the vertex so parabola

opens downwards $\therefore a$ is negative . Distance of directrix from

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

$= \frac{1}{4 \cdot 12.5} = \frac{1}{50} \therefore a = - \frac{1}{50}$.

Hence equation of parabola is $y = - \frac{1}{50} {\left(x - 20\right)}^{2} + 2.5$

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