To determine the equation of the parabola with the given conditions:
1. Identify the focus and directrix:
- The focus is [tex]\((4, 0)\)[/tex].
- The directrix is the vertical line [tex]\(x = -4\)[/tex].
2. Understand the definition of the parabola:
- The parabola is defined as the set of all points [tex]\((x, y)\)[/tex] that are equidistant from the focus and the directrix.
3. Set up the distance formula:
- The distance from a point [tex]\((x, y)\)[/tex] to the focus [tex]\((4, 0)\)[/tex] can be expressed using the distance formula:
[tex]\[
\sqrt{(x - 4)^2 + y^2}
\][/tex]
- The distance from a point [tex]\((x, y)\)[/tex] to the directrix [tex]\(x = -4\)[/tex] is:
[tex]\[
|x + 4|
\][/tex]
4. Equate the two distances:
According to the definition of the parabola, these distances should be equal:
[tex]\[
\sqrt{(x - 4)^2 + y^2} = |x + 4|
\][/tex]
5. Square both sides to eliminate the square root and absolute value:
[tex]\[
(x - 4)^2 + y^2 = (x + 4)^2
\][/tex]
6. Expand both sides:
[tex]\[
x^2 - 8x + 16 + y^2 = x^2 + 8x + 16
\][/tex]
7. Simplify the equation:
- Subtract [tex]\(x^2 + 16\)[/tex] from both sides:
[tex]\[
x^2 - 8x + 16 + y^2 - x^2 - 8x - 16 = 0
\][/tex]
- Combine like terms:
[tex]\[
y^2 - 16x = 0
\][/tex]
- Therefore, rearrange the terms to:
[tex]\[
y^2 = 16x
\][/tex]
Given this detailed solution, the equation that represents the parabola is:
[tex]\[
y^2 = 16x
\][/tex]
Thus, the correct choice is:
[tex]\[
\boxed{y^2 = 16x}
\][/tex]
