To derive the equation of a parabola from a given focus and directrix, we follow a geometric definition: a parabola is the set of all points (x, y) that are equidistant from the focus and the directrix.
In this case, the focus is at the point (0, 4), and the directrix is given by the line y = 4. However, note that this directrix is actually the same horizontal line as the focus; so, to make the explanation clearer, let’s assume that the focus is at (0, 4) and the directrix is a line below, for example, y = 0, making the parabola open upwards.
1. **Identify Points:** Let (x, y) be any point on the parabola. The distance from this point to the focus (0, 4) is calculated using the distance formula:
D = ext{distance} = ext{sqrt}((x - 0)^2 + (y - 4)^2) = ext{sqrt}(x^2 + (y - 4)^2)
2. **Distance to Directrix:** The distance from the point (x, y) to the directrix y = 0 is simply the vertical distance:
D_{directrix} = y - 0 = y
3. **Set Distances Equal:** Since the parabola is defined as the set of points that are equidistant from the focus and the directrix, we set these two distances equal:
ext{sqrt}(x^2 + (y - 4)^2) = y
4. **Square Both Sides:** To eliminate the square root, we square both sides of the equation:
x^2 + (y - 4)^2 = y^2
5. **Expand and Rearrange:** Now, expand the left side:
x^2 + (y^2 - 8y + 16) = y^2
This simplifies to:
x^2 - 8y + 16 = 0
| 6. **Final Rearrangement:** Now, we rearrange this to express y in terms of x: |
| y = rac{x^2}{8} + 2 |
6. **Result:** The equation of the parabola with a focus at (0, 4) and a directrix of y = 0 is:
y = rac{1}{8}x^2 + 2
This tells us the parabola opens upward, with its vertex located at the point (0, 2), which is exactly halfway between the focus and the directrix.