Question 1018042
<pre>
{{{9x^2-12x}}}{{{""=""}}}{{{36y+8}}}

Divide through by 9

{{{x^2-expr(12/9)x}}}{{{""=""}}}{{{4y+8/9}}}

{{{x^2-expr(4/3)x}}}{{{""=""}}}{{{4y+8/9}}}

Complete the square:

               {{{expr(-4/3)*expr(1/2)=-4/6=-2/3}}}
               {{{(-2/3)^2=4/9}}}

Add to both sides

{{{x^2-expr(4/3)x+4/9}}}{{{""=""}}}{{{4y+8/9+4/9}}}

{{{(x-2/3)^2}}}{{{""=""}}}{{{4y+12/9}}}

{{{(x-2/3)^2}}}{{{""=""}}}{{{4y+4/3}}}

{{{(x-2/3)^2}}}{{{""=""}}}{{{4(y+1/3)}}}

{{{(x-h)^2}}}{{{""=""}}}{{{4p(y-k)}}}

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

Vertex = (h,k) = {{{(matrix(1,3,2/3,",",-1/3))}}}

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 = {{{-1/3}}}, add p=1, get {{{2/3}}},

so focus = {{{(matrix(1,3,2/3,",",2/3))}}}.

x-coordinate of right endpoint of latus rectum = 
x-ccordinate of focus plus half of latus rectum's length,
2p, get {{{2/3+2(1)=2/3+6/3=8/3}}}
  
x-coordinate of left endpoint of latus rectum = 
x-ccordinate of focus minus half of latus rectum's length,
2p, get {{{2/3-2(1)=2/3-6/3=-4/3}}}

y-coordinates of ends of latus rectum = same as y-coordinate
of focus, {{{2/3}}}

right end of latus rectum = {{{(matrix(1,3,8/3,",",2/3))}}}

right end of latus rectum = {{{(matrix(1,3,-4/3,",",2/3))}}}

Directrix has equation y = y-coodinate of vertex minus p = 1
which is {{{-1/3-1=-1/3-3/3=-4/3}}}

Directrix has equation {{{y=-4/3}}}

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

{{{drawing(400,400,-4,5,-4,5, graph(400,400,-4,5,-4,5,x^2/4-x/3-2/9),

circle(0.66666667,-0.33333333,0.09),circle(0.66666667,-0.33333333,0.07),circle(0.66666667,-0.33333333,0.05),circle(0.66666667,-0.33333333,0.03),circle(0.66666667,-0.33333333,0.01),

circle(0.66666667,0.66666667,0.09),circle(0.66666667,0.66666667,0.07),circle(0.66666667,0.66666667,0.05),circle(0.66666667,0.66666667,0.03),circle(0.66666667,0.66666667,0.01),

blue(line(2/3-2,2/3,2/3+2,2/3)),






green(line(-5,-1/3-1,6,-1/3-1)) )}}}

Edwin</pre>