Identify the equation of a parabola whose focus is at (5,-1) and whose directrix is x=3.
vertex=(-5,-1)
p=-6
4p=-24
(x-5)=(-24)(y+1)^2
I got it wrong
Hint:
The focus is at (5,-1), and the directrix is at x=3, so the vertex lies halfway between the two.
The value of p is the distance between the vertex and the focus.
Wiki has an excellent article on the subject.
http://en.wikipedia.org/wiki/Parabola
To identify the equation of a parabola, you need to know the focus and the directrix. The standard equation of a parabola with its vertex at (h, k) is given by:
(x - h)^2 = 4p(y - k)
Where:
- (h, k) represents the coordinates of the vertex.
- p is the distance between the vertex and the focus (also the distance between the vertex and the directrix).
- The focus is represented by (h + p, k).
- The directrix is a vertical line with the equation x = h - p.
Given that the focus is at (5, -1) and the directrix is x = 3, we can determine the vertex and the value of p:
1. The vertex is the midpoint between the focus and the directrix. Thus, the x-coordinate of the vertex, h, is the average between 5 and 3: (5 + 3) / 2 = 4. The y-coordinate, k, will be the same as the focus since the parabola is symmetric.
Therefore, the vertex is (4, -1).
2. The distance between the vertex and the focus is p. Since the y-coordinates of both the vertex and the focus are the same (-1), the value of p is the absolute value of the difference between the x-coordinates of the vertex and the focus: |4 - 5| = 1.
Hence, p = 1.
Now we can substitute the values for the vertex and p into the standard equation:
(x - 4)^2 = 4(1)(y + 1)
Simplifying further:
(x - 4)^2 = 4y + 4
This is the equation of the parabola.