Focus is at #(-18,30) #and directrix is #y=22#. Vertex is at midway
between focus and directrix. Therefore vertex is at
#(-18,(30+22)/2)# i.e at #(-18, 26)# . The vertex form of equation
of parabola is #y=a(x-h)^2+k ; (h.k) ;# being vertex. Here
# h= -18 and k =26#. So the equation of parabola is
#y=a(x+18)^2 +26 #. Distance of vertex from directrix is
#d= 26-22=4#, we know # d = 1/(4|a|)#
#:. 4 = 1/(4|a|) or |a|= 1/(4*4)=1/16#. Here the directrix is below
the vertex , so parabola opens upward and #a# is positive.
#:. a=1/16# . The equation of parabola is #y=1/16(x+18)^2 +26 #
or #1/16(x+18)^2 = y-26 or (x+18)^2 = 16(y-26) # or
# (x+18)^2 = 4*4(y-26) #.The standard form is
#(x - h)^2 = 4p (y - k)#, where the focus is #(h, k + p)#
and the directrix is #y = k - p#. Hence the equation
of parabola in standard form is # (x+18)^2 = 16(y-26) #
graph{1/16(x+18)^2+26 [-160, 160, -80, 80]}
