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Magazine Chapter 7: Problem 41
Find an equation of a parabola that satisfies the given conditions. Focus \((0,-3)\) and directrix \(y=3\)
Short Answer
Expert verified
The equation of the parabola is \(x^2 = -12y\).
Step by step solution
01
Understand the Problem
We are given a focus of the parabola at \((0, -3)\) and a directrix of \(y = 3\). We need to find the equation of a parabola with these properties.
02
Identify the Vertex
The vertex of a parabola lies exactly midway between the focus and the directrix. Since the directrix is \(y = 3\) and the focus is \((0, -3)\), the vertex is halfway between, at \(y = 0\). The x-coordinate remains the same as the focus, so the vertex is \((0, 0)\).
03
Determine the Parabola Type
The parabola opens either upward or downward. Since the focus \((0, -3)\) is below the directrix \(y = 3\), the parabola opens downward.
04
Use the Standard Form for Vertical Parabola
The formula for a parabola that opens upwards or downwards with vertex at \((h, k)\) is \((x - h)^2 = 4p(y - k)\). The vertex is \((0, 0)\), so \(h = 0\) and \(k = 0\), simplifying to \(x^2 = 4py\).
05
Find the Value of p
The distance between the vertex and the focus is \(|p|\). Here, the focus is at \((0, -3)\) and vertex at \((0, 0)\), giving \(|p| = 3\) and since the parabola opens downwards, \(p = -3\).
06
Write the Equation
Substitute \(p = -3\) into the formula \(x^2 = 4py\). This gives \(x^2 = 4(-3)y\), or \(x^2 = -12y\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Focus and Directrix
The focus and directrix are fundamental elements in defining a parabola. The focus is a fixed point, and in this case, it's located at (0, -3). Every point on the parabola is equidistant from the focus and the directrix. The directrix, on the other hand, is a fixed line. Here, it's given by the equation \(y = 3\). To imagine a parabola, one must understand that it "curves" away, always maintaining equal distance from both the focus and the directrix.
This balance is key to forming the "U" shape of a parabola. So, essentially,
This balance is key to forming the "U" shape of a parabola. So, essentially,
- If you pick any point on the parabola, the distance to the focus is equal to the distance to the directrix.
- This property helps in deriving the equation of the parabola.
Vertex of a Parabola
The vertex is a crucial point on a parabola, representing its peak or lowest point. It's found right in the middle between the focus and directrix. Here, the directrix is \(y = 3\) and the focus is at (0, -3), so the vertex is at (0, 0). This middle point is where the parabola changes direction.
The x-coordinate of the vertex often equals the x-coordinate of the focus unless otherwise stated, simplifying calculations. In this example:
The x-coordinate of the vertex often equals the x-coordinate of the focus unless otherwise stated, simplifying calculations. In this example:
- The y-coordinate is the average of the focus and directrix: \[(0 - 3) + 3 = 0\]
- So, the vertex is directly above (or below in a different scenario) the focus following the axis of symmetry.
Standard Form of a Parabola
The standard form of the equation for a parabola is key to understanding its shape and position. For a parabola that opens vertically (upwards or downwards), the equation is
\((x - h)^2 = 4p(y - k)\).
Here, \((h, k)\) is the vertex's coordinates, and \(p\) is the distance from the vertex to the focus along the axis of symmetry. In the problem we're examining:
\((x - h)^2 = 4p(y - k)\).
Here, \((h, k)\) is the vertex's coordinates, and \(p\) is the distance from the vertex to the focus along the axis of symmetry. In the problem we're examining:
- The vertex is at \((0, 0)\), so the equation simplifies to \(x^2 = 4py\).
- The focus distance \(|p|\) is 3, since the focus is at (0, -3) relative to the vertex. So, \(p = -3\) because the parabola opens downward.
Parabola Orientation
A parabola's orientation tells us which direction it opens. It can be upward, downward, leftward, or rightward. Our parabola has a vertical orientation, meaning it goes upwards or downwards. This orientation is determined by the relative positions of the focus and directrix.
In this case, the focus at (0, -3) lies beneath the directrix (y = 3), guiding the parabola to open downward.
In this case, the focus at (0, -3) lies beneath the directrix (y = 3), guiding the parabola to open downward.
- An upward opening would require the focus to be above the directrix.
- If a horizontal orientation were considered, the focus and directrix would affect the x-coordinates rather than y.
