To find the equation of the parabola given the focus and directrix, we start by identifying the key characteristics. The focus of the parabola is at the point (4, 0), and the directrix is the line x = 4.
The general form for a parabola that opens horizontally is given by the equation (y – k)² = 4p(x – h), where (h, k) is the vertex of the parabola, and p is the distance from the vertex to the focus or the directrix.
In this case, since the focus is at (4, 0), the vertex must lie on the axis of symmetry—which in this case is a vertical line through the focus—so the x-coordinate (h) of the vertex is also 4. The directrix x = 4 indicates that the vertex is at the midpoint between the directrix and the focus. Thus, the vertex is at (4, 0).
The distance from the vertex to the focus is 0, and since the focus and directrix are both vertical, we can conclude that the parabola opens horizontally away from the directrix. Therefore, we can determine that p = 0.
Putting this information together, the vertex of this parabola is at (4, 0), and since p = 0, the equation of the parabola simplifies to:
(y – 0)² = 4 * 0 * (x – 4)
Which simplifies further to show that the equation of the parabola is:
y² = 0
This means that the parabola is a point at (4, 0) and does not open up or down with any width because the focus and directrix are coincident.