equation for parabola with focus (8,-2) directrix x=4
The standard parabola y^2 = 4px has focus at (p,0). So, we want to make that distance=4, so p=2.
That gives us y^2 = 8x for a parabola with focus at (8,0). Now shift that focus to (8,4) and we have
(y-4)^2 = 8x
To find the equation of a parabola given its focus and directrix, we can use the geometric definition of a parabola. A parabola is defined as the set of all points that are equidistant to the focus and directrix.
1. First, let's find the vertex of the parabola. Since the directrix is a vertical line given by x = 4, the vertex will lie on the line halfway between the focus and directrix. Hence, the x-coordinate of the vertex is the average of the x-coordinate of the focus and the directrix, which is (8 + 4) / 2 = 12 / 2 = 6.
So, the vertex is (6, y).
2. Now, let's calculate the distance between the focus and directrix. Since the directrix is a vertical line, the distance is the difference between the x-coordinate of the focus and the x-coordinate of the directrix. In this case, the distance is 8 - 4 = 4.
3. The distance between any point (x, y) on the parabola and the focus is called the focal length (f). Similarly, the distance between the point (x, y) and the directrix is the perpendicular distance (p). Using these values, we can determine the equation of the parabola.
4. Since the focus is (8, -2), the distance from the focus to any point (x, y) on the parabola is given by:
√((x - 8)² + (y + 2)²) = f
5. Since the directrix is x = 4, the perpendicular distance from any point (x, y) on the parabola to the directrix is given by:
|x - 4| = p
6. According to the geometric definition of a parabola, the focal length (f) is equal to the perpendicular distance (p). As we found earlier, f = p = 4.
7. Now we can use the vertex form of the equation of a parabola, which is:
(x - h)² = 4p(y - k)
In this case, the vertex is (6, y) and p = 4.
So, the equation becomes:
(x - 6)² = 4(4)(y - y)
= 16(y - y)
8. Simplifying the equation, we get:
(x - 6)² = 64(y - y)
(x - 6)² = 64y
Therefore, the equation of the parabola with focus (8, -2) and directrix x = 4 is (x - 6)² = 64y.