# What is the vertex form of the equation of the parabola with a focus at (2,-29) and a directrix of y=-23?

Aug 4, 2017

The equation of parabola is $y = - \frac{1}{12} {\left(x - 2\right)}^{2} - 26$ .

#### Explanation:

Focus of the parabola is $\left(2 , - 29\right)$

Diretrix is $y = - 23$ . Vertex is equidistant from focus and directrix

and rests at midway between them. So Vertex is at

$\left(2 , \frac{- 29 - 23}{2}\right)$ i.e at $\left(2 , - 26\right)$ . The equation of parabola in

vertex form is y= a(x-h)^2+k ; (h,k)  being vertex . Hence the

equation of parabola is $y = a {\left(x - 2\right)}^{2} - 26$ . The focus is below

the vertex so parabola opens downward and $a$ is negative here.

The distance of directrix from vertex is $d = \left(26 - 23\right) = 3$ and we

know $d = \frac{1}{4 | a |} \mathmr{and} | a | = \frac{1}{4 \cdot 3} = \frac{1}{12} \mathmr{and} a = - \frac{1}{12}$ Therefore,

the equation of parabola is $y = - \frac{1}{12} {\left(x - 2\right)}^{2} - 26$ .

graph{-1/12(x-2)^2-26 [-160, 160, -80, 80]} [Ans]