SOLUTION: Identify the focus, directrix, and axis of symmetry of the parabola 6x2+3y=06x2+3y=0.
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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 .
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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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