Problem 35 - Algebra and Trigonometry

Question

Find an equation for the parabola that has its vertex at the origin and satisfies the given condition(s). Opens upward with focus 5 units from the vertex

Step-by-Step Solution

Answer

The equation is \(x^2 = 20y\).

Step 1: Understand the properties of the parabola

A parabola with its vertex at the origin has an equation of the form \(y^2 = 4px\) if it opens right or left, and \(x^2 = 4py\) if it opens up or down. Since this parabola opens upward, the equation will be \(x^2 = 4py\).

Step 2: Identify the condition given

We are given that the parabola has a focus 5 units from the vertex. For a parabola opening upward with vertex at the origin, the focus lies at \((0, p)\). Thus, \(p = 5\).

Step 3: Plug the value of p into the parabola equation

Substitute \(p = 5\) into the equation \(x^2 = 4py\). This yields \(x^2 = 4(5)y = 20y\). So, the equation of the parabola is \(x^2 = 20y\).

Key Concepts

Vertex of a Parabola

The vertex of a parabola is a crucial point that essentially acts as the "turning point" of the parabola. It marks the point where the parabola changes direction. Whether the parabola opens upward, downward, left, or right, the vertex provides vital information about its position and orientation.
The standard equation for a parabola that opens upward (or downward) is given by \[ x^2 = 4py \] where \((0, 0)\) represents the vertex when the parabola is centered at the origin. In real-world scenarios, the vertex could lie at any point \((h, k)\). If this happens, the equation transforms into:\[ (x - h)^2 = 4p(y - k) \].
From this formula, it's apparent that when the vertex is at the origin, as in the given exercise, the equation becomes simpler with \(h = k = 0\). Hence, it simplifies our process of finding the parabola's equation.

Focus of a Parabola

The focus of a parabola is a fixed point that, together with the directrix, defines the parabola. It's important because every point on the parabola is equidistant to both its focus and its directrix.
For an upward or downward opening parabola with its vertex at the origin, the focus point is located at \((0, p)\) where \(p\) is the distance from the vertex to the focus. This distance can help determine the "width" or "sharpness" of the parabola. In the context of the problem, we know that the parabola's focus is 5 units away from the vertex. Thus, \(p = 5\). This information plugs directly into the parabola's equation, which is \( x^2 = 4py \). Substituting \(p = 5\) results in \( x^2 = 20y \).
Understanding where the focus lies helps to visualize how "opened" the parabola is based on the value of \(p\). The larger \(p\) is, the wider the parabola.

Upward Opening Parabola

An upward opening parabola is one where the "arms" of the parabola extend upward, forming a "U" shape. It's essential to use the correct form of the parabola equation to represent this configuration.
The general form for a parabola opening upward is \[ x^2 = 4py \] and this equation tells us several things. First, the square on \(x\) indicates symmetry around the \(y\)-axis. The term \(4py\) informs us of how wide or narrow the parabola will be based on the value of \(p\). A larger \(p\) makes the parabola wider, while a smaller \(p\) makes it narrower. In the provided exercise, since the parabolas open upward and the vertex is at the origin, we plug in \(p = 5\) to finally achieve the equation:\[ x^2 = 20y \].
Recognizing whether a parabola opens upwards or in another direction is essential for solving various geometry and physics-related problems. Identifying these characteristics quickly and accurately opens the door to deeper mathematical analysis.