To determine the equation of the parabola with the given vertex and focus, let's start by understanding the placement and properties of the parabola.
Given:
- The vertex of the parabola is at [tex]\((0,0)\)[/tex].
- The focus of the parabola is at [tex]\(\left(\frac{1}{8}, 0\right)\)[/tex].
Since the focus [tex]\(\left(\frac{1}{8}, 0\right)\)[/tex] lies on the x-axis and to the right of the vertex [tex]\((0,0)\)[/tex], this indicates that the parabola opens horizontally along the x-axis. The general form for a horizontally opening parabola with its vertex at the origin [tex]\((0,0)\)[/tex] is given by [tex]\(x = a y^2\)[/tex].
The given coordinates of the focus help us identify the value of the parameter [tex]\(p\)[/tex], the distance from the vertex to the focus. Here, the distance [tex]\(p\)[/tex] is:
[tex]\[ p = \frac{1}{8} \][/tex]
For a parabola opening horizontally, the relationship between [tex]\(a\)[/tex] and [tex]\(p\)[/tex] is given by:
[tex]\[ a = \frac{1}{4p} \][/tex]
Substituting the given value of [tex]\(p\)[/tex]:
[tex]\[ a = \frac{1}{4 \times \frac{1}{8}} = \frac{1}{\frac{1}{2}} = 2 \][/tex]
Therefore, the equation of the parabola in its standard form is:
[tex]\[ x = 2 y^2 \][/tex]
Among the given options:
A. [tex]\(x = 2 y^2\)[/tex]
B. [tex]\(y = -2 x^2\)[/tex]
C. [tex]\(x = -2 y^2\)[/tex]
D. [tex]\(y = 2 x^2\)[/tex]
The correct equation is:
[tex]\[ \boxed{x=2 y^2} \][/tex]
Thus, the correct answer is:
A. [tex]\(x = 2 y^2\)[/tex]
