Problem 20 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 19-30: Find an equation of the parabola that satisfies the given conditions. Focus \(F(0,-4)\) directrix \(y=4\)

Step-by-Step Solution

Answer

The equation of the parabola is \( x^2 = -16y \).

Step 1: Understand the Problem

We need to find the equation of a parabola given the focus and directrix. The focus is \( F(0, -4) \), and the directrix is \( y = 4 \). A parabola is defined as the set of all points that are equidistant from the focus and the directrix.

Step 2: Calculate Vertex

The vertex of the parabola is the midpoint between the focus and the directrix. Since the focus is \( (0, -4) \) and the directrix is \( y = 4 \), the vertex is halfway at \( (0, 0) \).

Step 3: Identify Parabola's Orientation

Since the vertex is at \( (0, 0) \), with the focus below and the directrix above, the parabola opens downwards. This is because the focus is below the directrix.

Step 4: Use the Standard Form of Parabola Equation

For a vertical parabola centered at \( (h, k) \) with equation \( (x - h)^2 = 4p(y - k) \), we have \( h = 0 \) and \( k = 0 \). The value \( p \) is the distance from the vertex to the focus, \( p = -4 \). So the equation becomes \( x^2 = 4(-4)y \) or \( x^2 = -16y \).

Step 5: Write Final Equation

Combining all the information from the previous steps, the equation of the parabola is \( x^2 = -16y \).

Key Concepts

Focus and Directrix

The focus and directrix are fundamental concepts when dealing with parabolas. Imagine the focus as a point and the directrix as a line. A parabola is formed by all points that are equally distant from both the focus and the directrix. This unique property defines its characteristic U-shape.

• Focus: The given focus was at point \( F(0, -4) \). This basically tells you that for the parabola in consideration, the significant focal point lies on the y-axis
  and below the vertex.
  
• Directrix: The directrix given as \( y = 4 \) is a horizontal line. It is situated above the focus in this problem.
  

Understanding these elements helps in identifying key features of the parabola's shape and position.

Vertex of a Parabola

In parabolas, the vertex is the point where the parabola changes direction and it serves as the midpoint between the focus and the directrix. For this exercise, you found the vertex by calculating the midpoint of the focus and the directrix.

The steps to find the vertex:

• Focus: \( F(0, -4) \)
• Directrix: \( y = 4 \)

So, you take the average of these two points to find the vertex
which sits nicely at \( (0, 0) \). This calculation is simple yet essential for defining the parabola.

Orientation of a Parabola

Understanding the orientation of a parabola is crucial for sketching its graph. The orientation tells us if the parabola opens upwards, downwards, left, or right.

In this example, the focus is beneath the vertex, while the directrix is above it. This configuration leads to a downward-facing parabola, as it will open towards the focus.

• Focus below the directrix: opens downwards.
• Vertex: at the origin \( (0, 0) \).

The direction of the parabola is dictated by the focus and directrix positions. Always remember, the parabola curves toward the focus.

Standard Form of Parabola

The standard form of a parabola's equation helps in easily identifying its features. Generally, for a parabola symmetric about the y-axis, it is expressed as:

• \((x - h)^2 = 4p(y - k)\)
• Here, \((h, k)\) represent the vertex.
• \(p\) is the distance from the vertex to the focus.

For the given problem, you have:

• Vertex \((h, k): (0,0)\)
• \(p = -4\) since the distance from the vertex \((0,0)\) to the focus \((0,-4)\) is 4 in the downward direction.

Using all these, you form the parabola's equation as
\(x^2 = 4(-4)y\) which simplifies to \(x^2 = -16y\).
This equation captures the parabola's opened-downward orientation, vertex location, and shape.