To determine the equation of a parabola that opens to the right with a vertex at the origin [tex]\((0,0)\)[/tex] and a focus at [tex]\((9,0)\)[/tex], we follow these steps:
1. Identify the general form of the equation for a parabola opening to the right. Such a parabola has the equation:
[tex]\[ x = ay^2 \][/tex]
where [tex]\(a\)[/tex] is a constant related to the distance from the vertex to the focus.
2. Determine the parameter [tex]\(a\)[/tex]. For a parabola opening to the right with vertex at the origin and focus at [tex]\((f,0)\)[/tex], the constant [tex]\(a\)[/tex] is given by:
[tex]\[ a = \frac{1}{4f} \][/tex]
3. Given the focus is at [tex]\((9,0)\)[/tex], we have [tex]\(f = 9\)[/tex].
4. Substitute [tex]\(f = 9\)[/tex] into the formula for [tex]\(a\)[/tex]:
[tex]\[ a = \frac{1}{4 \cdot 9} = \frac{1}{36} \][/tex]
5. Substitute [tex]\(a = \frac{1}{36}\)[/tex] back into the general parabolic equation:
[tex]\[ x = \frac{1}{36} y^2 \][/tex]
Therefore, the equation that represents a parabola opening to the right with a vertex at the origin and a focus at [tex]\((9,0)\)[/tex] is:
[tex]\[ x = \frac{1}{36} y^2 \][/tex]
So, the correct answer is:
[tex]\[ x = \frac{1}{36} y^2 \][/tex]
