# What is the equation of the parabola with a focus at (3,18) and a directrix of y= 23?

Jun 18, 2017

Equation of parabola is $y = - \frac{1}{10} {\left(x - 3\right)}^{2} + 20.5$

#### Explanation:

Focus at $\left(3 , 18\right)$ and directrix of $y = 23$.

Vertex is at equidistant from focus and directrix.

So vertex is at $\left(3 , 20.5\right)$ . The distance of directrix from vertex is d= 23-20.5=2.5 ; d = 1/(4|a|) or 2.5 = 1/(4|a|) or a = 1/(4*2.5)=1/10

Since directrix is above vertex , the parabola opens downwards and $a$ is negative. So $a = - \frac{1}{10} , h = 3 , k = 20.5$

Hence equation of parabola is $y = a {\left(x - h\right)}^{2} + k \mathmr{and} y = - \frac{1}{10} {\left(x - 3\right)}^{2} + 20.5$

graph{-1/10(x-3)^2+20.5 [-80, 80, -40, 40]} [Ans]