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Chapter 10: Problem 29

Find an equation for the parabola that has its vertex at the origin and satisfies the given condition(s). Directrix \(x=2\)

Short Answer

Expert verified
The equation of the parabola is \(y^2 = -8x\).

Step by step solution

01

Understand the Parabola's Orientation

Since the directrix is vertical (\(x = 2\)), the parabola opens horizontally. Specifically, this parabola opens to the left or right.
02

Use the Vertex Form of a Parabola

For a parabola that opens horizontally with its vertex at the origin, the equation is \((y - k)^2 = 4p(x - h)\). Here, \((h, k) = (0, 0)\) since the vertex is at the origin.
03

Determine the Value of \(p\)

The directrix is at \(x = 2\). For a horizontally opening parabola, the directrix is \(x = h - p\). Since \(h = 0\), we get \(0 - p = 2\), hence \(p = -2\).
04

Substitute the Values into the Vertex Form

Substitute \(k = 0\), \(h = 0\), and \(p = -2\) into the vertex form equation: \((y - 0)^2 = 4(-2)(x - 0)\).
05

Simplify the Equation

Simplifying the equation yields \(y^2 = -8x\). This is the equation of the parabola with vertex at the origin and directrix at \(x = 2\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vertex Form of a Parabola
The vertex form of a parabola equation is a helpful way to express parabolas that reveals the location of their vertex, as well as important information about their orientation. For horizontally oriented parabolas, the vertex form looks like this:
  • \((y - k)^2 = 4p(x - h)\)
In this expression of the equation:
  • \(h\) and \(k\) are the coordinates of the vertex \((h, k)\).
  • \(p\) denotes the distance between the vertex and the focus, as well as the vertex and the directrix. The value of \(p\) determines the direction the parabola opens. If \(p > 0\), the parabola opens to the right, and if \(p < 0\), it opens to the left.
In our exercise, the vertex is at the origin, which simplifies our equation to:
  • \(y^2 = 4p(x)\)
This form makes it easier to substitute the necessary values to find the specific equation of any parabola with its vertex at the origin.
Directrix and Focus Relationship
A parabola is uniquely defined by its relationship with its focus and directrix. These two elements are crucial in understanding how the parabola is structured. The focus is a point from which distances are measured to form the shape of the parabola, while the directrix is a line serving as a reference for these distances.
  • The equation for a parabola can be derived by knowing the location of either, the directrix or the focus. For a horizontally oriented parabola, the directrix is given by \(x = h - p\).
  • In our problem, the directrix is \(x = 2\). Since the vertex is at the origin \((0, 0)\), and substituting \(h = 0\), we find that \(-p = 2\). Therefore, \(p = -2\).
This shows the horizontal distance between the vertex and the directrix. The focus would then be located at \((h + p, k) = (-2, 0)\) since the parabola opens horizontally to the left.
Horizontal Parabola Orientation
Parabolas come in different orientations depending on their axis of symmetry, which gives rise to the terms 'vertical' or 'horizontal' parabola. Knowing the direction in which a parabola opens is crucial for writing its accurate equation.
  • A horizontally oriented parabola opens either to the left or the right. This orientation is indicated by the directrix being vertical as opposed to horizontal.
  • In our context, given that the directrix is \(x = 2\), the parabola opens horizontally since the directrix is parallel to the x-axis.
As established, the orientation is determined by \(p\), where \(p < 0\) implies a leftward opening. Hence, for the exercise, \(p = -2\) means our parabola will open leftwards from its vertex at the origin.This overall relationship between the vertex form, focus, and directrix underpins the breathing design of the parabola, allowing us to predict and calculate its exact shape and equation.

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