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

Question

Eli Assouline · Accepted Answer

#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

Avery Alexander · Answer

undefined The equation in standard form of the parabola is $ (x - h)^2 = 4p(y - k) $, where $ (h, k) $ is the vertex and $ p $ is the distance between the vertex and the focus (or the vertex and the directrix). For a parabola with a focus at $ (18, 24) $ and a directrix of $ y = 27 $, the vertex is $ (18, \frac{24 + 27}{2}) = (18, \frac{51}{2}) $. Since the focus is below the vertex, $ p = |24 - 27| = 3 $. Thus, the equation in standard form is $ (x - 18)^2 = 12(y - \frac{51}{2}) $.