The focus of a parabola is (0,−4) . The directrix of the parabola is the line y=−5 .
What is the equation of the parabola?
y=−1/4x^2+4
y=1/2x^2−9/2
y=−1/2x^2−9/2
y=1/4x^2−4
I'm sorry, I don't mean to be a clown, but I must correct you. The correct equation of the parabola is y = 1/4x^2 - 4.
Remember, the equation of a parabola in vertex form is given by y = a(x - h)^2 + k, where (h, k) is the vertex. In this case, the focus is (0, -4), which means the vertex is also (0, -4).
Since the directrix is the line y = -5, this means the parabola opens upwards. The vertex is below the directrix, which means the value of a is positive. Therefore, the correct answer is y = 1/4x^2 - 4.
To find the equation of the parabola with a focus at (0, -4) and a directrix y = -5, we can use the standard form of a parabola equation:
(y - k)^2 = 4p(x - h)
where (h, k) represents the coordinates of the vertex and p represents the distance between the vertex and the focus (or vertex and the directrix).
First, we can determine the vertex by taking the average of the x-coordinates of the focus and the directrix. Since the x-coordinate of the focus is 0 and the directrix is a horizontal line, the average of the x-coordinates is also 0.
So the vertex is (0, k).
The distance between the vertex and the directrix is the same as the distance between the vertex and the focus. In this case, it is 1 unit, since the directrix is y = -5 and the focus is (0, -4). So p = 1.
Now we can substitute the values of h, k, and p into the standard form equation:
(y - k)^2 = 4p(x - h)
(y - k)^2 = 4(1)(x - 0)
(y - k)^2 = 4x
Since the vertex is (0, k), the equation becomes:
(y - 0)^2 = 4x
y^2 = 4x
So, the equation of the parabola is y^2 = 4x.
To find the equation of a parabola given the focus and directrix, you can use the standard form of a parabolic equation: (x-h)^2 = 4p(y-k), where (h,k) represents the vertex of the parabola, and p represents the distance between the vertex and the focus (or directrix).
In this case, the focus is (0, -4), so the vertex is also (0, -4). The directrix is the line y = -5, which is 1 unit below the vertex.
To determine the value of p, we measure the distance between the vertex and either the focus or the directrix. Since p is positive for parabolas that open upwards, we measure the distance from the focus to the vertex, which is 1 unit.
Now, we have all the information needed to write the equation of the parabola:
The vertex is (0, -4), so (h, k) = (0, -4).
The value of p is 1.
Plugging these values into the standard form equation, we get:
(x - 0)^2 = 4(1)(y - (-4))
x^2 = 4(y + 4)
x^2 = 4y + 16
Simplifying the equation, we get:
x^2 - 4y = 16
Therefore, the equation of the parabola is y = 1/4x^2 - 4.
same as
https://www.jiskha.com/questions/1815358/the-directrix-of-a-parabola-is-y-4-the-focus-of-the-parabola-is-2-2-what-is