To derive the equation of the parabola with a given focus and directrix, we can use the definition of a parabola: it is the set of all points that are equidistant from the focus and the directrix.
In this case, we have:
- Focus (F): (0, 5)
- Directrix (D): y = 5
The first step is to understand that a parabola opens either upwards or downwards. Since the focus is directly above the directrix, this parabola will open upwards.
Let (x, y) be a point on the parabola. The distance from this point to the focus and to the directrix must be equal:
- Distance to the focus: √((x – 0)² + (y – 5)²)
- Distance to the directrix: |y – 5|
Setting these two distances equal gives us:
√((x - 0)² + (y - 5)²) = |y - 5|Since the parabola opens upwards, we have that y – 5 is non-negative, allowing us to replace |y – 5| with (y – 5):
√((x)² + (y - 5)²) = (y - 5)Next, we square both sides to eliminate the square root:
(x)² + (y - 5)² = (y - 5)²Upon simplifying this, we subtract (y – 5)² from both sides:
(x)² = 0From this equation, we find:
x² = 0Thus, the equation of the parabola is:
y = 0 (for all values of y)In summary, the equation of the parabola whose focus is at (0, 5) and directrix at y = 5 is:
(x - 0)² = 4p(y - k)Here, p is the distance from the vertex to the focus, which is 0 in this case. Therefore, we can express it simply as:
(x - 0)² = 0So the parabola basically converges at the line where the focus and the directrix are equal.