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Chapter 7: Problem 258

For the following exercises, determine the equation of the parabola using the information given. Focus (2,3) and directrix \(x=-2\)

Short Answer

Expert verified
The equation of the parabola is \(x^2 = 8y - 24\).

Step by step solution

01

Understand the Problem

We are given the focus of a parabola at (2,3) and a directrix of \(x=-2\). The parabola's equation can be derived using these elements as it opens horizontally due to the vertical directrix.
02

Use the Definition of a Parabola

A parabola is defined as the set of points equidistant from the focus and the directrix. So, we use the formula \((x-h)^2 = 4p(y-k)\) for a parabola with a horizontal orientation, where \((h,k)\) is the vertex, determined from the focus and directrix.
03

Find the Vertex

The vertex's x-coordinate lies midway between the x-coordinate of the focus and the directrix line. Since the directrix \(x=-2\) and the focus is at \(x=2\), the midpoint is \(x=0\). Thus, the vertex is \((0, 3)\).
04

Determine the Parameter p

The parameter \(p\) is the distance from the vertex to the focus along the x-axis. Since the focus is at \(x=2\) and the vertex at \(x=0\), \(p = 2\).
05

Write the Parabola Equation

Using the vertex \((0,3)\) and \(p=2\), we substitute into the standard equation \((x-h)^2=4p(y-k)\). Thus, \((x-0)^2=8(y-3)\) simplifies to \(x^2=8y-24\).

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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 in understanding the geometry of a parabola. Think of the focus as a point and the directrix as a straight line that guide the shape of the parabola.
  • The parabola is the set of all points that are equidistant from these two elements.
  • If you take any point on the parabola, its distance to the focus is equal to its distance to the directrix.
To illustrate, let's consider a parabola with a focus at point directrix as a vertical line, say, at some x-coordinate. Here, the directrix is given as \(x = -2\).The importance of focus and directrix lies in defining the parabola's geometry accurately.
Horizontal Parabola
A horizontal parabola opens to the left or right depending on the location of its focus and directrix.
  • In this context, the focus is positioned along the horizontal axis from the directrix.
  • Our example has a focus at \((2, 3)\) and a directrix at \(x = -2\), resulting in a horizontally opening parabola.
This orientation is defined by the parabolic equation's format: \((x-h)^2 = 4p(y-k)\).Here, the variable \(x\) is squared, signifying a horizontal arrangement, unlike the vertical arrangement where \(y\) would be squared.
Vertex of a Parabola
The vertex is the turning point of the parabola and its coordinates are crucial in forming the equation.
  • It lies exactly halfway between the focus and the directrix along the axis of symmetry of the parabola.
  • In the problem provided, the vertex is calculated as halfway between the focus鈥檚 point on the x-axis and the directrix, giving us \((0,3)\).
Therefore, the vertex is not only a coordinate but also a pivotal component in determining the parabola鈥檚 equation, ensuring that it is positioned correctly in the coordinate plane.
Standard Form of Parabola
The standard form for a horizontal parabola's equation provides a systematic way to establish its shape and direction.
  • The equation is structured as \((x-h)^2 = 4p(y-k)\), where \((h,k)\) is the vertex.
  • In this equation, \(4p\) represents the distance between the vertex and the focus or the vertex and the directrix.
Using our example of a vertex at \((0,3)\) and a distance (or \(p\)) of 2, the specific equation becomes \((x-0)^2 = 8(y-3)\).This illustrates how the standard form allows us to easily represent and graph any parabola, guiding students to solve similar problems effectively.

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Most popular questions from this chapter

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