Understanding the Parabola
A parabola is a U-shaped curve that can open upwards, downwards, left, or right. It is defined as the set of points that are equidistant from a point called the focus and a line known as the directrix.
Finding the Equation
To derive the equation of a parabola from its focus and directrix, follow these steps:
1. Identify the Focus and Directrix
Assume the focus of the parabola is at point (h, k) and the equation of the directrix is y = d (if the parabola opens vertically) or x = d (if it opens horizontally). Here, d represents the y or x-coordinate of the directrix, depending on the orientation of the parabola.
2. Determine the Orientation
If the parabola opens upwards or downwards, use:
(x - h)² = 4p(y - k)
If the parabola opens to the right or left, the equation changes to:
(y - k)² = 4p(x - h)
3. Calculate p
The variable p is the distance from the focus to the directrix. If the focus is above the directrix, p is positive; if below, p is negative.
4. Complete the Equation
Substituting the value of p into the chosen equation based on the orientation will give you the final equation of the parabola.
Example
Say the focus is at (2, 3) and the directrix is the line y = 1. Here’s how you would write the equation:
- Focus:
(h, k) = (2, 3) - Directrix:
y = 1impliesd = 1 - Distance
p=3 - 1 = 2 - The parabola opens upwards, so we use:
(x - 2)² = 4 * 2 * (y - 3)
This simplifies to:
(x - 2)² = 8(y - 3)
Conclusion
By following these steps, you can write the equation of a parabola using its focus and directrix. Just remember to identify the orientation and find the distance p correctly!