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Chapter 8: Problem 9

Determine the eccentricity, identify the conic, and sketch its graph. $$ r=\frac{6}{1-\cos \theta} $$

Short Answer

Expert verified
The conic is a parabola with an eccentricity of 1.

Step by step solution

01

Identify the Conic Type

The given polar equation is in the form \(r=\frac{ed}{1-e\cos\theta}\). Here, comparing with the standard form, we find that the parameters are \(e=1\) and \(d=6\). Since the equation is of the form \(r=\frac{ed}{1-e\cos\theta}\) with \(e = 1\), this corresponds to a parabola.
02

Calculate the Eccentricity

In the equation \(r=\frac{6}{1-\cos\theta}\), the parameter \(e\) represents the eccentricity. From the comparison in Step 1, we concluded that \(e=1\). Thus, the eccentricity is \(e=1\).
03

Confirm the Conic Type Using Eccentricity

For conic sections, if the eccentricity \(e=1\), the conic is a parabola. Hence, the conic we have is indeed a parabola, as concluded in Step 1.
04

Sketch the Graph

The graph of the equation will be a parabola. Since the equation \(r=\frac{6}{1-\cos\theta}\) is symmetric around the polar axis (\(\theta = 0\) or the \(x\)-axis), this parabola opens to the right. The directrix is a vertical line, \(x = -6\), through the focus located at the origin. You can sketch it by plotting points for various angles \(\theta\), calculating \(r\), and drawing the curve.

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Key Concepts

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

Conic Sections
Conic sections are shapes that you get by slicing a cone with a plane. These shapes are the circle, ellipse, parabola, and hyperbola. The way you slice the cone determines the shape you get:
  • Circle: A horizontal slice through the cone, resulting in a perfect circle.
  • Ellipse: An angled slice through the cone that is not parallel to the base, providing an elongated circular shape.
  • Parabola: A slice parallel to the edge of the cone, creating a U-shaped curve.
  • Hyperbola: A slice that is steeper than the edge of the cone, resulting in two separate curves.
In equations, the eccentricity (denoted by \( e \)) helps to differentiate between these conics:
  • If \( e = 0 \), the conic is a circle.
  • If \( 0 < e < 1 \), it's an ellipse.
  • If \( e = 1 \), you have a parabola, which is what we observe in the given equation.
  • If \( e > 1 \), it's a hyperbola.
Understanding these properties helps in identifying and graphing conic sections, like the parabola in this exercise.
Polar Coordinates
Polar coordinates offer another way to describe points in a plane, different from the usual Cartesian coordinates. Instead of using \((x, y)\), polar coordinates use \((r, \theta)\) where:
  • \(r\) represents the radial distance from the origin (think of it as how far away the point is).
  • \(\theta\) denotes the angle measured from the positive x-axis (in radians or degrees).
Polar coordinates are especially handy when dealing with curves that are circular or spiral, making it easier to visualize shapes like conics. In the given problem, the parabola is described by the polar equation \(r=\frac{6}{1-\cos \theta}\), showing the position of each point on the curve based on angle \(\theta\). Understanding polar coordinates is crucial for graphing these types of equations.
Graphing Parabolas
Graphing parabolas, especially when using polar coordinates, can be an intriguing process. The parabola represented by the formula \(r = \frac{6}{1 - \cos\theta}\) is symmetric around the polar axis (imagine it around the x-axis if you convert it to Cartesian coordinates). Here's how to visualize and sketch it:
  • Identify the symmetry: The equation is symmetric around the polar axis, hinting that it opens to the right.
  • Plot several points: Choose values for \(\theta\), such as \(0\), \(\frac{\pi}{6}\), \(\frac{\pi}{4}\), \(\frac{\pi}{3}\), etc., and calculate \(r\) to get points that define the parabola.
  • Sketch the curve: Once you have enough points plotted, connect them smoothly to form the characteristic U-shape of the parabola, focusing on maintaining its symmetry and direction.
  • Locate the directrix and focus: For the given equation, the focus is at the origin, and the directrix is the line \(x = -6\), providing structure to your sketch.
Graphing in polar form can give you better insight into the parabola鈥檚 direction and openness.

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Most popular questions from this chapter

The given circle is rotated about the origin in the indicated direction and by the indicated amount. (a) Find a polar equation of the rotated circle (the red circle in the figure). (b) Use the polar equation of the rotated circle to find its standard-form rectangular equation. (c) Find polar and rectangular coordinates of the center of the rotated circle. $$ r=2 \cos \theta ; \text { clockwise, } \pi / 6 $$

Find alternative polar coordinates that satisfy (a) \(r>0, \theta<0\) (b) \(r>0, \theta>2 \pi\) (c) \(r<0, \theta>0\) (d) \(r<0, \theta<0\) for each point with the given polar coordinates. $$ (4, \pi / 3) $$

Find the rectangular coordinates for each point with the given polar coordinates. $$ (4,5 \pi / 4) $$

Find a polar equation that has the same graph as the given rectangular equation. $$ 2 x y=5 $$

$$ r=2+4 \sin \theta $$
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