A parabola can be drawn given a focus of (5,−10) and a directrix of y=6. What can be said about the parabola?
we have p = -8, so
(x-5)^2 = -32(y+2)
I apologize, as an AI language model, I made a mistake in my previous response. You are correct, since the parabola opens downwards, the value of p is negative. Therefore, the equation of the parabola can be written as:
(x - 5)^2 = -32(y + 2)
This equation is in vertex form, with vertex at (5, -2) and p = -8.
To understand what can be said about the parabola, we first need to understand what a focus and a directrix are.
A parabola is a symmetric curve that can be defined as the set of all points that are equidistant to a fixed point called the focus and a fixed line called the directrix. The focus and the directrix play key roles in determining the shape and position of the parabola.
In this case, the given focus is (5, -10) and the given directrix is y = 6. Now let's analyze what we can infer about the shape and position of the parabola based on this information:
1. Orientation: Since the directrix is a horizontal line (y = 6) and the focus is not on the directrix, we can infer that the parabola is vertically oriented. This means the parabola will open either upwards or downwards.
2. Vertex: The vertex of the parabola is the point equidistant to both the focus and the directrix. In this case, the vertex will be the midpoint between the focus and the directrix. To find the vertex, we can use the formula V = (h, k), where h is the x-coordinate of the focus and k is the y-coordinate of the directrix.
V = (5, (6 + (-10)) / 2) = (5, -2)
3. Axis of Symmetry: The axis of symmetry is a vertical line that passes through the vertex and divides the parabola into two symmetrical halves. The equation for the axis of symmetry is x = h, where h is the x-coordinate of the vertex. In this case, the axis of symmetry will be x = 5.
4. Focus and Directrix relationship:
- The focus lies on the axis of symmetry, in this case, at (5, -10).
- The distance from any point on the parabola to the focus is equal to the distance from that point to the directrix. This property is known as the "focus-directrix property."
With this information, we can conclude that the parabola's axis of symmetry is vertical, opening either upward or downward, with the vertex at (5, -2), the focus at (5, -10), and the directrix at y = 6.
- The vertex of the parabola is halfway between the focus and the directrix, which is at (5, -2).
- The parabola will open downwards because the directrix is above the focus.
- The distance between the vertex and the focus (or the directrix) is called the "p" value of the parabola. In this case, the distance between the vertex and the directrix is 8 units, so the p value is -8 (since the parabola opens downwards). Therefore, the equation of the parabola can be written as: (y + 2) = -1/8(x - 5)^2.