SOLUTION: write an equation of a parabola with a vertex (3,0) and a directix at x = 3. I get that this information gives you a = 0 and this does not make sense.
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-> SOLUTION: write an equation of a parabola with a vertex (3,0) and a directix at x = 3. I get that this information gives you a = 0 and this does not make sense.
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Question 198607: write an equation of a parabola with a vertex (3,0) and a directix at x = 3. I get that this information gives you a = 0 and this does not make sense. Answer by solver91311(24713) (Show Source):
It actually does make a perverse kind of sense. What you have is a degenerate parabola, actually a straight line that isn't a parabola at all. Here's why.
Since the given directrix is a vertical line, the axis of symmetry of this parabola is horizontal. The vertex form of the equation of a parabola with vertex at (h,k) and directrix x = h - p is:
So using the equation of the directrix to calculate p when (h,k) = (3, 0) and x = 3:
So the equation becomes:
Which is the equation of the x-axis.
Geometrically speaking, a parabola is formed by the intersection of a plane and a cone where the plane is parallel to one of the generators of the cone. But if you move the plane to a point where the generator of the cone actually lies wholly in the plane, the parabola degenerates to a straight line. The distance from the focus to the vertex (and the distance from the directrix to the vertex) is a function of the distance from the axis of symmetry that lies in the plane and the generator of the cone. When that distance is zero, you have no parabola.
