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Chapter 10: Problem 22

Determine whether the statement is true or false. Explain your answer. If \(d\) is a positive constant, then the conic section with polar equation $$r=\frac{d}{1+\cos \theta}$$ is a parabola.

Short Answer

Expert verified
The statement is true; the conic is a parabola since the eccentricity is 1.

Step by step solution

01

Identify the Conic Type

The given polar equation is \( r=\frac{d}{1+\cos \theta} \). This equation has the general form \( r=\frac{ed}{1+e\cos \theta} \) where \( e \) is the eccentricity of the conic. The standard form equation represents different types of conic sections depending on the value of \( e \): a circle if \( e=0 \), an ellipse if \( 01 \). The given equation resembles a parabola equation in polar coordinates because the eccentricity \( e \) is equal to 1.
02

Derive the Eccentricity

To determine the eccentricity, match the given equation \( r=\frac{d}{1+\cos \theta} \) to the standard form \( r=\frac{ed}{1+e\cos \theta} \). Notice that in the given equation's formula, the expression in the denominator, \( 1 + \cos \theta \), suggests an eccentricity \( e = 1 \). This confirms that the conic section is a parabola since \( e=1 \) in the standard form represents a parabola.
03

Conclusion on the Conic

Since the derived value of the eccentricity \( e \) is 1, the conic section described by the polar equation \( r=\frac{d}{1+\cos \theta} \) is indeed a parabola, as per the properties of conic sections defined by their eccentricities.

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Key Concepts

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

Polar Coordinates
In mathematics, polar coordinates offer a different perspective for visualizing points on a plane. Unlike Cartesian coordinates, which use a grid of x and y values, polar coordinates are based on a point's distance from a central point (the origin) and the angle from a reference direction, usually the positive x-axis.

This system has immense applications in understanding curves and shapes, like conic sections, in a much intuitive way:

  • Point Location: Every point in the polar coordinate system is defined by a pair \( r, \theta \), where \( r \) is the radial distance from the origin, and \( \theta \) is the angle measured counterclockwise from the positive x-axis.
  • Converting Polar Equations: Polar equations can often be converted into more familiar Cartesian formats, which helps in visualizing and solving complex problems.
  • Simplifying Conics: Polar coordinates simplify finding and analyzing conic sections, as seen when identifying an equation like \( r=\frac{d}{1+\cos \theta} \). This can help in visualizing how conic sections behave and change under different conditions.
Understanding and using polar coordinates is essential for solving complex geometric problems, especially those involving conic structures.
Eccentricity
Eccentricity (\( e \)) is a crucial concept for understanding the shape of conic sections. It determines how "stretched" a conic section is compared to a circle, where every point is equidistant from a center.

Here's a brief overview of what the eccentricity signifies for various conics:

  • Circle: When \( e=0 \), the conic is a perfect circle with no deviation.
  • Ellipse: In the case of \( 0 < e < 1 \), the shape is an ellipse, which is elongated but still closed.
  • Parabola: An eccentricity of \( e=1 \), as with the given polar equation \( r=\frac{d}{1+\cos \theta} \), forms a parabola which is open and no longer circles back on itself.
  • Hyperbola: If \( e > 1 \), the conic becomes a hyperbola, signifying a significant stretching that results in two separate curves.
The role of eccentricity is pivotal in elliptical and parabolic orbits, as it dictates the path objects like planets or satellites follow around larger bodies.
Parabola
A parabola is a distinct type of conic section identified by its unique geometric properties. It's an open curve with several important characteristics:

  • Defining Feature: A parabola is known because it has an eccentricity (\( e \)) of exactly 1. This makes it differ from other conics like ellipses and hyperbolas, which have a wider eccentricity range.
  • Focus and Directrix: Every parabola is defined by its focus鈥攁 point鈥攁nd a directrix鈥攁 line. The parabola is the locus of points equidistant from both the focus and the directrix.
  • Symmetrical Nature: Parabolas have an axis of symmetry, which is a line that passes through the vertex, the highest or lowest point of the curve.
  • Real-World Applications: From satellite dishes to reflective properties of headlights, parabolas play a crucial role in many practical applications, enforcing their significance in geometry and physics.
By understanding what defines a parabola, students can better predict and analyze the trajectory of objects and the reflective properties these shapes might impose.

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Most popular questions from this chapter

Sketch the ellipse, and label the foci, vertices, and ends of the minor axis. (a) \(9 x^{2}+4 y^{2}-18 x+24 y+9=0\) (b) \(5 x^{2}+9 y^{2}+20 x-54 y=-56\)

Show that in a polar coordinate system the distance \(d\) between the points \(\left(r_{1}, \theta_{1}\right)\) and \(\left(r_{2}, \theta_{2}\right)\) is $$d=\sqrt{r_{1}^{2}+r_{2}^{2}-2 r_{1} r_{2} \cos \left(\theta_{1}-\theta_{2}\right)}$$

Sketch the ellipse, and label the foci, vertices, and ends of the minor axis. (a) \(\frac{x^{2}}{25}+\frac{y^{2}}{4}=1\) (b) \(4 x^{2}+y^{2}=36\)

Sketch the parabola, and label the focus, vertex, and directrix. (a) \(y^{2}=4 x\) (b) \(x^{2}=-8 y\)

Find an equation for a hyperbola that satisfies the given conditions.\([\) Note: In some cases there may be more than one hyperbola. \(]\) (a) Vertices (0,6) and (6,6)\(;\) foci 10 units apart. (b) Asymptotes \(y=x-2\) and \(y=-x+4\) passes through the origin.
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