SOLUTION: Find the vertex and focus and end points of the latus rectum of the parabola 9x^2-12x-36y-8=0

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Question 1018042: Find the vertex and focus and end points of the latus rectum of the parabola 9x^2-12x-36y-8=0
Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
9x%5E2-12x%22%22=%22%2236y%2B8

Divide through by 9

x%5E2-expr%2812%2F9%29x%22%22=%22%224y%2B8%2F9

x%5E2-expr%284%2F3%29x%22%22=%22%224y%2B8%2F9

Complete the square:

               expr%28-4%2F3%29%2Aexpr%281%2F2%29=-4%2F6=-2%2F3
               %28-2%2F3%29%5E2=4%2F9

Add to both sides

x%5E2-expr%284%2F3%29x%2B4%2F9%22%22=%22%224y%2B8%2F9%2B4%2F9

%28x-2%2F3%29%5E2%22%22=%22%224y%2B12%2F9

%28x-2%2F3%29%5E2%22%22=%22%224y%2B4%2F3

%28x-2%2F3%29%5E2%22%22=%22%224%28y%2B1%2F3%29

%28x-h%29%5E2%22%22=%22%224p%28y-k%29

h=2, 4p=4, p=1, k=-1/3

Vertex = (h,k) = %28matrix%281%2C3%2C2%2F3%2C%22%2C%22%2C-1%2F3%29%29

Length of latus rectum = 4p = 4.
                          p = 1.

Distance from vertex to the focus = 

Distance from vertex to the directrix = p = 1

y-coordinate of vertex = -1%2F3, add p=1, get 2%2F3,

so focus = %28matrix%281%2C3%2C2%2F3%2C%22%2C%22%2C2%2F3%29%29.

x-coordinate of right endpoint of latus rectum = 
x-ccordinate of focus plus half of latus rectum's length,
2p, get 2%2F3%2B2%281%29=2%2F3%2B6%2F3=8%2F3
  
x-coordinate of left endpoint of latus rectum = 
x-ccordinate of focus minus half of latus rectum's length,
2p, get 2%2F3-2%281%29=2%2F3-6%2F3=-4%2F3

y-coordinates of ends of latus rectum = same as y-coordinate
of focus, 2%2F3

right end of latus rectum = %28matrix%281%2C3%2C8%2F3%2C%22%2C%22%2C2%2F3%29%29

right end of latus rectum = %28matrix%281%2C3%2C-4%2F3%2C%22%2C%22%2C2%2F3%29%29

Directrix has equation y = y-coodinate of vertex minus p = 1
which is -1%2F3-1=-1%2F3-3%2F3=-4%2F3

Directrix has equation y=-4%2F3

blue line segment is latus rectum
green line is directrix
Upper point marked is focus
Lower point marked is vertex



Edwin