Question 1061806: Identify the focus, directrix, and axis of symmetry of the parabola 6x2+3y=06x2+3y=0.
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website! I believe the equation you tried to write is  .
<--> <--> <--> .
THE EXPECTED APPROACH:
You have memorized a few "facts" and "formulas," including the ones listed below.
1) If the equation can be written in the form 
for some real numbers  ,  , and  , with  ,
it is the equation of a parabola whose axis of symmetry is the "vertical" line  (parallel to the y-axis),
and is the x-coordinate of the vertex of the parabola.
2) For a parabola with a "vertical" axis of symmetry,
you can find the y-coordinate,  , of the vertex of the parabola,
by substituting  for  in the equation of the parabola.
3) For a parabola with a "vertical" axis of symmetry,
if  , and is the coordinate of the vertex,
the coordinates of the focus are
 ,
and the directrix is the line  .
Applying that to , we have  ,  ,
 .
So ,  is the axis of symmetry and the x-coordinate of the vertex.
The y-coordinate of the vertex is  .
The coordinates of the focus are  ,
and the equation of the directrix is  .
THE NON-MEMORIZER APPROACH:
You realize that is shows x as an even function of x
(one that assigns the same y value to any pair of opposite values of x, and  ),
meaning that the y-axis,  , is the axis of symmetry.
You realize that  , making (0,0) a maximum and the vertex of the parabola.
That also tells you that the focus is a the focal distance  below the vertex at  ,
and that the directrix is the horizontal line  ,
both at the same distance from the vertex, and on opposite sides of the vertex.
You may remember that  ,
or you may use the definition of parabola to find .
All points is a parabola are at the same distance from focus and directrix.
The point with  , at a distance  from the focus,
has the y-coordinate  .
Its distance to directrix is
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