Question

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

Answer 1

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

Explanation:

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]