Problem 31 - Algebra and Trigonometry

Question

Find an equation for the parabola that has its vertex at the origin and satisfies the given condition(s). Directrix \(y=-10\)

Step-by-Step Solution

Answer

The equation of the parabola is \(x^2 = 40y\).

Step 1: Understanding the Problem

We need to find an equation for a parabola that has its vertex at the origin \((0,0)\) and whose directrix is the line \(y = -10\). The vertex form of a parabola equation is key in solving the problem.

Step 2: Vertex Form of the Parabola

With the vertex at the origin, the equation of the parabola can be expressed in the form \(x^2 = 4py\) where \(p\) is the distance from the vertex to the focus or from the vertex to the directrix. We need to determine \(p\).

Step 3: Finding the Value of \(p\)

The vertex \((0,0)\) and the directrix \(y = -10\) imply that \(p = 10\) (the distance between the vertex and the directrix along the y-axis). Thus, the directrix is 10 units below the vertex.

Step 4: Parabola's Equation

With \(p = 10\), the equation becomes \(x^2 = 4(10)y\), simplifying to \(x^2 = 40y\). This is the equation of the parabola.

Key Concepts

Vertex Form of Parabola

The vertex form of a parabola is an essential tool in identifying the characteristics and equation of a parabola. This form is especially convenient when the vertex, or the highest or lowest point of the parabola, is known. The general expression for the vertex form is

• For a vertical parabola: \[ y = a(x - h)^2 + k \] where
  • \( (h, k) \) is the vertex
  • \( a \) determines the direction and width of the parabola
• For a horizontal parabola: \[ x = a(y - k)^2 + h \] where
  • \( (h, k) \) is again the vertex
  • \( a \) influences its opening and stretch

In our specific exercise, we have a vertical parabola with the vertex at the origin, which simplifies our equation to \[ x^2 = 4py \] where \( p \) is the key element.
The form \( x^2 = 4py \) highlights the symmetrical nature of the parabola around the y-axis, with the multiplication factor \( 4p \) demonstrating how the parabola scales. Understanding this structure is crucial in proceeding to work with directrices and foci related to parabolas.

Directrix of a Parabola

The directrix of a parabola is a fixed line used in the geometric definition of the parabola and plays a vital role in its algebraic description. The directrix helps in ascertaining the distance-related properties of the parabola.

• The concept of the directrix is coupled with the focus: the line from which the distance of any point on the parabola must equate to the distance from the focus.
• For a vertical parabola, if the vertex is at \( (0,0) \), and the directrix is specified, like in our case with \( y = -10 \), then the distance \( p \) from the vertex to the directrix is perpendicular along the y-axis.

Since in our example the parabola's directrix is given at \( y = -10 \), the distance \( p \) to the vertex at the origin \( (0,0) \) is \( 10 \) units, confirming the consistent relationship between the parabola's geometric definition and its algebraic equation.
Understanding the directrix aids in the comprehension of the entire structure and orientation of a parabola in a graph.

Focus of a Parabola

The focus of a parabola is equally important as the vertex or directrix and is part of the fundamental definition of a parabola in terms of distances. The focus is a point from which every point on the parabola is equidistant to as it is to the directrix. This point influences the shape and orientation of the parabola.

• For a vertical parabola, where the vertex is at \( (0,0) \), if the directrix \( y = -10 \) is known, the focus \( (0, p) \) should be placed symmetrically opposite of the vertex relative to the directrix.
• In our case, since \( p = 10 \), the focus will be at \( (0, 10) \), 10 units above the vertex along the y-axis.

This focus can be visualized as a guiding point for the "arms" of the parabola as they extend infinitely away from it.
By understanding the placement of the focus in relation to the vertex and directrix, students can easily understand how to draw and analyze the parabola's properties, making it easier to predict behavior like reflection or light focusing.