Write the equation of a parabola with the given focus and directrix Focus: (-2, 0) Directrix: x = 4
AAAaannndd the bot gets it wrong yet again!
Your parabola does not even open in the right direction!
If the directrix is x=4, then the parabola opens left or right -- its axis is perpendicular to the line x=4. Now, the focus, at (-2,0) is to the left of the line x=4, so the parabola opens to the left.
Now recall that the parabola y^2 = 4px has
focus at x = (p,0)
vertex at (0,0)
directrix x = -p
Our parabola has its directrix 6 units to the right of the focus, so p = -3, and the vertex is at (1,0)
The axis of symmetry is the line y = 0, so the equation is
y^2 = -12(x-1)
to verify, click on the "properties" button at
wolframalpha. com/input?i=y%5E2+%3D+-12%28x-1%29
I apologize for my mistake in the previous response. You are correct that the parabola should open to the left because the focus is to the left of the directrix.
Using the formula you provided:
- Focus: (-2, 0)
- Directrix: x = 4
We know that the vertex will be the midpoint between the focus and the directrix.
Vertex: ((-2+4)/2, 0) = (1, 0)
The distance between the focus and the vertex is 3 units (same as the distance between the directrix and the vertex), so the value of p is -3 (negative because the focus is to the left of the vertex).
Therefore, the equation of the parabola is:
(y - 0)^2 = -12(x - 1)
which simplifies to:
y^2 = -12(x - 1)
Thank you for correcting my mistake.
To write the equation of a parabola with a given focus and directrix, you can follow these steps:
Step 1: Determine whether the parabola is vertical or horizontal. In this case, the directrix is a vertical line (x = 4), so the parabola is vertical.
Step 2: Find the vertex of the parabola. The vertex is the midpoint between the focus and the directrix. In this case, the vertex is at (-2, 2).
Step 3: Find the distance between the vertex and the focus (also known as the focal length). In this case, the focal length is |-2 - 4| = 6.
Step 4: Use the vertex and focal length to determine the equation of the parabola. The equation of a vertical parabola with vertex (h, k) and focal length p is given by:
(x - h)^2 = 4p(y - k)
In this case, the vertex is (-2, 0) and the focal length is 6.
Plugging these values into the equation, we get:
(x - (-2))^2 = 4(6)(y - 0)
Simplifying further, we have:
(x + 2)^2 = 24y
Therefore, the equation of the parabola with the given focus (-2, 0) and directrix x = 4 is (x + 2)^2 = 24y.
Since the directrix is a vertical line, we know that the parabola opens horizontally.
The vertex of the parabola will be the midpoint between the focus and the directrix.
Vertex: (1, 0)
Since the focus is 3 units to the left of the vertex, we know that the equation of the parabola is of the form:
(x - h)^2 = 4p(y - k)
where (h, k) is the vertex, and p is the distance between the vertex and the focus.
In this case:
h = 1
k = 0
p = 2
Substitute these values into the equation:
(x - 1)^2 = 8y
This is the equation of the parabola with focus (-2, 0) and directrix x = 4.