write an equation of parabola with vertex at the origin and the given directix

Question

directrix y=1.4

Answer 1

To understand directrix we need to know what a parabola is.  A parabola is a set of points that is equidistant between a point and a line.  This means that the distance between a point on a parabola and the focus is the same as the distance between that same point on the parabola and the directrix. The point is a focus, the main point.  The line is the directrix (horizontal line). 

 

One point on the parabola that is equidistant between the focus and the directrix is the vertex of the parabola.

 

Vertex = (0, 0)

Directrix = (0, 1.4)

 

Lets find the focus.  The x coordinates among them are the same.  The coordinates are different.

 

y_vertex = (y_focus + y_directrx) / 2

 

 

(2 * y_vertex) - y_directrix = y_foucs

 

(2 * 0) - 1.4 = y_focus

 

-1.4 = y_focus

 

 

The focus is (0, -1.4).

 

 

Now, if we draw this graphically and apply the above concepts., we can see that the parabola will open downward.  What the directrix does is tells us what direction the parabola will open depending on other given points.

 

The vertex form of the equation of the parabola is

 

y = -(x - 0)²  + 0

 

y = -x²