# What is the equation of the parabola with a focus at (3,18) and a directrix of y= 17?

Jun 5, 2017

The equation of parabola is $y = \frac{1}{2} {\left(x - 3\right)}^{2} + 17.5$

#### Explanation:

The focus is at $\left(3 , 18\right)$ .Directrix is $y = 17$

The vertex is at equidistant from focus and directrix. So vertex is at $\left(3 , 17.5\right)$. Distance of directrix from vertex is $d = 0.5 \therefore a = \frac{1}{4 d} = \frac{1}{4 \cdot \frac{1}{2}} = \frac{1}{2}$

The standard equation of parabola is y=a(x-h)^2+k ; (h,k) being vertex. Here the directrix is behind vertex, so parabola opens upward and $a$ is positive.

So the equation of parabola is $y = \frac{1}{2} {\left(x - 3\right)}^{2} + 17.5$
graph{1/2(x-3)^2+17.5 [-80, 80, -40, 40]} [Ans]