fill out the equation of a parabola with the focus of (-2,3) and the directrix is x=4
y^2 = 4px has
vertex at (0,0)
focus at (p,0)
directrix at x = -p
The distance from focus to directrix is thus 2p, and the vertex is at (1,3)
So the parabola must be
(y-3)^2 = -12(x-1)
see the graph at
https://www.wolframalpha.com/input/?i=parabola+%28y-3%29%5E2+%3D+-12%28x-1%29
To fill out the equation of a parabola with a given focus and directrix, we can use the standard form:
(x - h)^2 = 4p(y - k)
where (h, k) is the coordinates of the vertex, and p is the distance between the vertex and the focus (also the distance between the vertex and the directrix).
In this case, the focus is (-2, 3) and the directrix is x = 4.
First, let's find the vertex by taking the average of the x-coordinates of the focus and the directrix. The x-coordinate of the vertex will be:
(h) = (-2 + 4) / 2 = 2 / 2 = 1
So, the vertex is at (1, k).
Next, we need to find the distance between the vertex and either the focus or the directrix to determine the value of p.
The distance between the vertex (1, k) and the focus (-2, 3) is the same as the distance between the vertex and the directrix x = 4. This distance is:
| 1 - 4 | = 3
Since p is the distance between the vertex and the focus or the directrix, we find that p = 3.
Now we can substitute the values of the vertex (1, k) and p = 3 into the standard form:
(x - 1)^2 = 4(3)(y - k)
Simplifying further, we have:
(x - 1)^2 = 12(y - k)
This is the equation of the parabola in standard form with the focus (-2, 3) and directrix x = 4.
To fill out the equation of a parabola with the given focus and directrix, we first need to determine whether the parabola opens vertically or horizontally.
Given that the directrix is a vertical line (x = 4), the parabola will open horizontally. The axis of the parabola will be parallel to the directrix.
Next, we need to find the vertex of the parabola. The vertex lies halfway between the focus and the directrix, along the axis of the parabola.
Since the directrix is the vertical line x = 4 and the focus is (-2,3), the vertex will have coordinates [(4 + (-2))/2, 3] = [1,3].
Now we can use the vertex (1,3) and the focus (-2,3) to determine the distance between them, which is the same as the distance between the vertex and the directrix.
The distance between the vertex (1,3) and the directrix x = 4 is 4 - 1 = 3 units.
Knowing the distance between the vertex and the focus, we can write the equation of the parabola.
For a parabola opening horizontally with vertex (h, k), focus (a, k), and directrix x = (h - p), where p is the distance between the vertex and the focus, the equation is:
(x - h)² = 4p(y - k)
Plugging in the values we have:
(h, k) = (1, 3)
(a, k) = (-2, 3)
p = 3
The equation becomes:
(x - 1)² = 4(3)(y - 3)
Simplifying further:
(x - 1)² = 12(y - 3)
Therefore, the equation of the parabola is (x - 1)² = 12(y - 3).