What is an equation of a parabola with a vertex at the origin and directrix x=4.75?
A parabola is defined as the set of all points that are equidistant from a fixed point, called the focus, and a fixed line, called the directrix.
For a parabola with a vertex at the origin (0,0) and a directrix of x = 4.75, we can deduce that the focus is also on the x-axis but at an equal distance on the opposite side of the vertex.
The distance "p" from the vertex to the directrix (and the focus) is thus 4.75 units. Hence, the focus is at (-4.75, 0).
We can write the equation of a parabola with horizontal axis in the form:
\[ x = ay^2 \]
Where "a" is the value related to the distance between the vertex and the focus (and therefore the directrix).
Since the focus is 4.75 units away from the vertex, we have:
\[ 4p = 4.75 \]
\[ p = \frac{4.75}{4} \]
\[ p = 1.1875 \]
Therefore, a = 1/(4p) = 1/(4 * 1.1875) = 1/4.75 = 0.21052631579
Thus, the equation of the parabla is:
\[ x = 0.21052631579y^2 \]
We can also express this more simply as
\[ x = \frac{1}{4.75}y^2 \]
This gives us a parabola that opens to the left (since the focus is to the left of the vertex) with a vertex at the origin and a directrix of x = 4.75.