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Chapter 10: Problem 281

Identify the conic with a focus at the origin, and then give the directrix and eccentricity. $$r=\frac{4}{7+2 \cos \theta}$$

Short Answer

Expert verified
The conic is a parabola with a directrix at \( d = 2 \) and eccentricity \( e = 1 \).

Step by step solution

01

Identify the Conic Section Type

The given equation is of the form \( r = \frac{ed}{1 + e \cos \theta} \), where \(e\) is the eccentricity, \(d\) is the distance from the focus to the directrix, and the conic section is determined by \(e\). Compare the given equation \( r = \frac{4}{7 + 2 \cos \theta} \) with the standard form and rearrange it to identify \(e\).
02

Rewrite the Equation to Identify Parameters

Rewrite the given equation to match the standard form: \( r = \frac{4}{7 + 2 \cos \theta} = \frac{4}{2 \left(\frac{7}{2} + \cos \theta\right)} = \frac{2}{\frac{7}{2} + \cos \theta}\). This reveals that \(e\cos\theta\) should match \(\cos \theta\), giving that \(e = 1\) and \( ed = 2 \).
03

Determine the Eccentricity

From the rewritten form \( r = \frac{2}{\frac{7}{2} + \cos\theta} \), we previously identified \(e = 1\). When \(e = 1\), the conic is a parabola, since a parabola has an eccentricity of exactly 1.
04

Find the Directrix

For a parabola \(r = \frac{ed}{1 + e\cos\theta} \), since we know \(ed = 2\) and \(e = 1\), it follows that \(d = 2\). Thus, the directrix is a line defined by this value and depends on the orientation, which is consistent with the given equation.

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Key Concepts

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

Eccentricity
Eccentricity, denoted as \( e \), is a number that describes how much a conic section deviates from being circular. It determines the shape of the conic section:
  • If \( e = 0 \), the conic is a circle, which is perfectly round.
  • If \( 0 < e < 1 \), the conic is an ellipse, slightly oval shaped.
  • If \( e = 1 \), the conic is a parabola, a curve that gets closer to a straight line at infinity.
  • If \( e > 1 \), the conic is a hyperbola, a shape with two opposite curves.
In the given exercise, the eccentricity is found to be \( e = 1 \), classifying the conic as a parabola.Knowing how to identify the eccentricity is essential to understanding the nature of the conic section you're dealing with. It's like having a map where each unique \( e \) value guides you to a different geometric shape. This understanding allows you to comprehend the geometry and symmetry involved in the given scenario.
Focus and Directrix
The focus and directrix are critical components in defining conic sections. They help establish the properties and orientation of the conic.
  • The focus is a fixed point at which the conic section is defined to maintain a constant relation to it through the eccentricity.
  • The directrix is a fixed line, and the conic section maintains a specific relationship to this line via its distance.
For parabolas, this relationship can be visualized by imagining any point on the conic section where the ratio of its distance to the focus and to the directrix is constant, defined by \( e = 1 \). In our example, the parabola's directrix is found using the equation \( r = \frac{2}{\frac{7}{2} + \cos\theta} \) to yield \( d = 2 \), where \( d \) is the perpendicular distance to the directrix. Thus, the directrix is a line chosen in such a way that it complements the parabola, ensuring its opening conforms to the recognized pattern.
Polar Coordinates
Polar coordinates are a way of representing points in a circular grid, distinct from the traditional rectangular coordinate system.
  • A point in polar coordinates is represented as \( (r, \theta) \).
  • \( r \) is the radial distance from the origin (often referred to as the focus in conics), and \( \theta \) is the angle formed with the positive x-axis.
They are especially useful when dealing with equations of circles, ellipses, parabolas, and hyperbolas, as they simplify the expression of curves that radiate around a center point. Within the context of the given problem, the use of polar coordinates makes it easier to express the relationship between points on the conic and the focus as well as the angle formed with the directrix. This form of representation allows for a more intuitive understanding of the geometry, focusing on radial distances and angles rather than mere x and y locations.

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