Question

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

Answer 1

Equation of parabola is `y = -1/10(x-3)^2+20.5`

Explanation:

Focus at `(3,18)` and directrix of `y=23`.

Vertex is at equidistant from focus and directrix.

So vertex is at `(3,20.5)` . 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=-1/10, h=3, k=20.5`

Hence equation of parabola is `y=a(x-h)^2+k or y = -1/10(x-3)^2+20.5`

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