How do you find the focus, vertex, and directrix of #x^2 -6x -36y +225=0#?

1 Answer
Narad T. · Andy Y.
Nov 3, 2016

The vertex is #(a,b)# #=(3,6)#
The focus is #(a,b+p/2)# #=(3,15)#
And the directrix is #y=b-p/2# ; #y=6-9=-3#

Explanation:

Let's rewrite the equation
#x^2-6x-36y+225=0#
#x^2-6x+9=36y-225+9#
#(x-3)^2=36y-216=36(y-6)#
This is the equation of a parabola
#(x-a)^2=2p(y-b)#
#a=3#
#b=6#
#p=18#
The vertex is #(a,b)# #=(3,6)#
The focus is #(a,b+p/2)# #=(3,15)#
And the directrix is #y=b-p/2# ; #y=6-9=-3#

graph{((y-(x-3)^2/36-6)(y+3))=0 [-41.1, 41.07, -20.53, 20.6]}

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