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

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Answer 1 · 2017-08-04T11:21:27

The equation of parabola is #y= -1/12(x-2)^2-26 # .

Focus of the parabola is # (2 , -29) #

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

and rests at midway between them. So Vertex is at

#( 2 , (-29-23)/2) # i.e at # (2 , -26)# . 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(x-2)^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= (26-23) =3 # and we

know #d = 1/(4|a|)or |a| =1/(4*3) =1/12 or a = -1/12 # Therefore,

the equation of parabola is #y= -1/12(x-2)^2-26 # .

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