What is the equation in standard form of the parabola with a focus at (18,24) and a directrix of y= 27?

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Answer 1 · 2017-08-27T21:38:04

$y = -1/6x^2+6x- 57/2 larr$ standard form

We know that the standard form for the equation of a parabola with a horizontal directrix is

$y = ax^2+bx+c$

but, because we are given the focus and the equation of the directrix, it is easier to start with the corresponding vertex form

$y = a(x-h)^2+k" [1]"$

and then convert to standard form.

We know that the x coordinate, "h", of the vertex is the same as the x coordinate of the focus:

$h = 18$

Substitute into equation [1]:

$y = a(x-18)^2+k" [2]"$

We know that the y coordinate, "k", of the vertex is the midpoint between the focus and the directrix:

$k = (24+27)/2$

$k = 51/2$

Substitute into equation [2]:

$y = a(x-18)^2+51/2" [3]"$

The focal distance, "f", is the signed vertical distance from the vertex to the focus:

$f = 24-51/2$

$f = -3/2$

We know that $a = 1/(4f)$

$a = 1/(4(-3/2)$

$a = -1/6$

Substitute into equation [3]:

$y = -1/6(x-18)^2+51/2$

Expand the square:

$y = -1/6(x^2-36x+ 324)+51/2$

Distribute the $-1/6$ :

$y = -1/6x^2+6x- 54+51/2$

Combine like terms:

$y = -1/6x^2+6x- 57/2 larr$ standard form