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Chapter 12: Problem 18

Exer. \(17-26:\) Find an equation for the conic that satisfies the given conditions. parabola, with focus \(F(-4,0)\) and directrix \(x=4\)

Short Answer

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

Step by step solution

01

Understanding the Parabola

A parabola is defined as the set of all points that are equidistant from a fixed point called the focus and a line called the directrix. In this problem, the focus is given as \( F(-4,0) \) and the directrix is the vertical line \( x = 4 \).
02

Identifying Vertex and Orientation

The vertex of the parabola lies halfway between the focus and the directrix. Since the focus is \( x = -4 \) and the directrix is \( x = 4 \), the vertex, which is also on the x-axis (same y-coordinate as the focus), is at \( x = 0 \). Thus, the vertex is at \( V(0, 0) \). The parabola opens to the left (towards the focus from the directrix).
03

Applying the Distance Formula

For a parabola with a horizontal axis, the standard form is \( (y-k)^2 = 4p(x-h) \), where \((h,k)\) is the vertex, and \( p \) is the distance from the vertex to the focus. Here, \((h,k)\) is \((0,0)\) and \( p = -4 \) due to the equation \( -4 - 0 = -4 \). This indicates the parabola opens to the left.
04

Writing the Equation

Substitute \( h = 0 \), \( k = 0 \), and \( p = -4 \) into the standard form \( (y-k)^2 = 4p(x-h) \). The equation simplifies to \( y^2 = -16x \). This represents the parabola with the given focus and directrix.

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Key Concepts

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

Focus of a Parabola
In the context of a parabola, the focus is crucial as it determines the shape and direction in which the parabola opens. The focus is a fixed point, and every point on the parabola is equidistant from this point and the directrix. For our example, the parabola has its focus at the point \( F(-4,0) \).
  • This tells us that the parabola's structure is defined around being equidistant from \( F(-4,0) \) and the line \( x = 4 \), known as the directrix.
  • Understanding how the focus relates to the rest of the parabola can help predict its trajectory and position.
The presence of a focus influences the orientation and the direct equations that can be formed from the geometric structure of the parabola.
Directrix
The directrix of a parabola serves as a line from which distances are measured to determine the curvature of the parabola. In this particular problem, the directrix is the vertical line \( x = 4 \).
  • Points on the parabola maintain equal distances from this line and the focus \( F(-4,0) \).
  • The directrix is always perpendicular to the axis of symmetry of the parabola, impacting where the parabola opens.
In the given example, the parabola opens to the left because the focus is to the left of the directrix, guiding the parabola towards \( x = -4 \).
Vertex of a Parabola
The vertex of a parabola is the point where the parabola changes direction. It is located exactly halfway between the focus and the directrix. For the given example, with the focus at \( F(-4,0) \) and the directrix at \( x = 4 \), the vertex is at the origin \( V(0, 0) \).
  • The vertex is also the maximum or minimum of the parabola, depending on its orientation.
  • In a coordinate plane, the vertex gives a central point from which the parabola symmetrically expands.
This central location at the origin simplifies calculations and allows for straightforward expressions when writing the parabola's equation.
Standard Form of a Parabola
The standard form of the equation for a parabola with a horizontal axis of symmetry is \( (y-k)^2 = 4p(x-h) \), where \((h, k)\) is the vertex and \(p\) is the distance from the vertex to the focus. In our example, the vertex is \((0,0)\), and since \(p = -4\) (because the focus is \(-4\) units away from the vertex), we substitute these into the standard form.
  • This leads to the equation \( y^2 = -16x \) for our specific parabola.
  • The negative sign of \(16x\) indicates the parabola opens to the left, as the value of \(p\) is negative.
Understanding and using the standard form helps in predicting the curve's direction and plotting the parabola accurately on a graph.

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