Write an equation of a parabola with the given focus and directrix. Focus: (0,3); directrix: y=-1
Since the directrix is a horizontal line, the parabola is vertical. The vertex is halfway between the focus and the directrix, which is at (0,1).
The distance between the focus and the vertex is the same as the distance between the vertex and the directrix, which is 4. This is also the value of the coefficient "a" in the standard form of the parabolic equation.
Therefore, the equation of the parabola is:
(x - 0)^2 = 4(y - 1)
or
x^2 = 4(y - 1)
are you sure?
I apologize, I made an error in my previous response. The correct equation of a parabola with the given focus and directrix is:
4p(y - k) = (x - h)^2
where (h,k) is the vertex, p is the distance between the vertex and the focus (and also the distance between the vertex and the directrix).
In this case, the vertex is (0,1) (as it is the midpoint between the focus and directrix). The focus is at (0,3) and the directrix is y = -1. The distance between the vertex and the focus (or directrix) is therefore 2 units.
Hence, we can plug in the values in the formula as:
4(2)(y - 1) = (x - 0)^2
8(y - 1) = x^2
Thus, the equation of the parabola with the given focus and directrix is x^2 = 8(y - 1).
are you sure
Yes, I am sure that the equation I provided is correct. It corresponds to a parabolic shape with a vertical axis of symmetry, whose vertex is at (0, 1), and whose focus is at (0, 3), and whose directrix is the horizontal line y = -1.
Another way to check that this equation describes the correct parabolic shape is to plot it on a graphing calculator or computer program and verify that it matches the given focus and directrix.
what would the work be if the answer was Y = 1/8x^2 + 1
If the equation of the parabola were Y = 1/8x^2 + 1, then its vertex would be (0, 1) and it would open upwards, given that the coefficient of x^2 (which determines the shape and orientation of the parabola) is positive.
To find the focus and directrix, we can use the formulae:
- "p" is the distance between the vertex and the focus (or directrix) along the axis of symmetry. For a parabola with a vertical axis, p is the coefficient of y in the equation, divided by 4 times the coefficient of x^2. In this case, p = 1/(4*1/8) = 2.
- The focus is located at the point (h, k+p), where (h, k) is the vertex. Therefore, the focus is at (0, 1+2) = (0,3).
- The directrix is a horizontal line located at a distance of "p" units below the vertex. Therefore, the directrix is y = 1-2 = -1.
We can check that this parabolic equation satisfies the given focus (0, 3) and directrix y = -1, which means that both equations describe the same parabolic shape.
is that correct
Yes, that's correct!
show your work
Certainly!
The given equation of the parabola is:
Y = 1/8x^2 + 1
This can be rewritten in standard form as:
8Y = x^2 + 8 (by multiplying both sides by 8)
This equation describes a parabola with a vertex at (0, 1) and a vertical axis of symmetry, since the coefficient of x^2 is positive and determines the shape and orientation of the parabola.
Next, we can use the formulae given above to find the focus and directrix:
- "p" is the distance between the vertex and the focus (or directrix) on the axis of symmetry. For a parabola with a vertical axis, p is equal to 1/(4a), where a is the coefficient of x^2 in the standard form. In this case, a = 1/8, so p = 2.
- The focus is located at the point (h, k+p), where (h, k) is the vertex. Therefore, the focus is at (0, 1+2) = (0,3).
- The directrix is a horizontal line located at a distance of "p" units below the vertex. Therefore, the directrix is y = 1-2 = -1.
Therefore, the equation Y = 1/8x^2 + 1 describes a parabola with a vertex at (0, 1), a focus at (0, 3), and a directrix at y = -1, which is consistent with the original problem statement.
so then why isnt that the answer
I apologize for the confusion. You are correct that the equation Y = 1/8x^2 + 1 describes the same parabolic curve with a vertex at (0,1), focus at (0,3), and directrix y=-1.
Both equations, x^2 = 8(y - 1) and Y = 1/8x^2 + 1, represent the same parabolic curve, and either equation could be used to describe the curve.
I apologize for any inconvenience caused by the error in my initial response, and I hope this clears up any confusion.