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
To identify the equation of a parabola, you need to know the coordinates of the focus and the equation of the directrix. Let's break down the process step by step.
1. The vertex of the parabola is given as (-5, -1).
2. The focus is at (5, -1), meaning the parabola opens towards the right (since the x-coordinate of the focus is greater than the x-coordinate of the vertex).
3. The directrix is given as x = 3, which is a vertical line parallel to the y-axis.
4. To find the equation of the parabola, you first need to find the distance between the vertex and the focus, which is known as the focal length (f). In this case, f = 5 - (-5) = 10.
5. The distance between the vertex and the directrix is equal to the focal length (f) as well.
6. Since the parabola opens to the right, the equation of the parabola can be written in the form (x - h)^2 = 4p(y - k), where (h, k) represents the vertex, and p represents the distance between the vertex and the focus (or directrix).
7. Substituting the vertex coordinates (-5, -1) into the equation, we get (x + 5)^2 = 4p(y + 1).
8. Now, we need to determine the value of p. Since p is the distance between the vertex and the focus or directrix, we can use the equation p = (x coordinate of the focus/directrix) - (x coordinate of the vertex). In this case, p = 5 - (-5) = 10.
9. Plugging this value into the equation from step 7, we obtain (x + 5)^2 = 4(10)(y + 1).
10. Simplifying further, we have (x + 5)^2 = 40(y + 1), which is the equation of the parabola.
So, the correct equation of the parabola with a focus at (5,-1) and a directrix of x = 3 is (x + 5)^2 = 40(y + 1).