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

1 Answer
Aug 3, 2017

Equation of parabola is #y = -1/50(x-20)^2+2.5#.

Explanation:

Focus is at # (20 ,-10)# and directrix is # y=15 #

Vertex is equidistant from focus and directrix and in midway

between them. So vertex is at # (20, (15-10)/2=2.5) or (20, 2.5)#

Equation of Parabola in vertex form is # y= a(x-h)^2+k# or

#y = a(x-20)^2+2.5# . Focus is below the vertex so parabola

opens downwards #:.a # is negative . Distance of directrix from

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

#= 1/(4*12.5)=1/50 :. a = - 1/50#.

Hence equation of parabola is #y = -1/50(x-20)^2+2.5#

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