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Chapter 10: Problem 50

Determine whether the statement is true or false. Explain your answer. The graph of a dimpled lima莽on passes through the polar origin.

Short Answer

Expert verified
False, a dimpled lima莽on does not pass through the polar origin since its minimum \(r\) value is greater than zero.

Step by step solution

01

Understanding the Lima莽on

A lima莽on is a type of polar graph given by the equation \( r = a + b \cos\theta \) or \( r = a + b \sin\theta \). The graph's shape depends on the values of \(a\) and \(b\). If \(|a| < |b|\), the lima莽on will be a limacon with an inner loop; if \(|a| = |b|\), it is a cardioid; and if \( |a| > |b| \), it is a dimpled lima莽on.
02

Condition for Passing Through the Polar Origin

For any polar graph \(r = f(\theta)\) to pass through the polar origin (or the point \(r = 0\)), there must exist some angle \(\theta\) such that \(f(\theta) = 0\). For a lima莽on given by \(r = a + b \cos\theta\) (or \(r = a + b \sin\theta\)), this means we need \(a + b\cos\theta = 0\) (or \(a + b\sin\theta = 0\)).
03

Analyzing the Dimpled Lima莽on

For a dimpled lima莽on where \(|a| > |b|\), we consider \(r = a + b\cos\theta\) (or \(r = a + b\sin\theta\)). The expression \(a + b\cos\theta\) ranges from \(a - |b|\) to \(a + |b|\) as \(\theta\) varies from 0 to \(2\pi\). Because \(|a| > |b|\), the minimum value \(a - |b|\) is greater than zero.
04

Conclusion

Since the minimum of \(r = a + b\cos\theta\) is greater than zero for a dimpled lima莽on, \(r\) never equals zero. Therefore, a dimpled lima莽on does not pass through the polar origin.

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Key Concepts

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

Polar Coordinates
Polar coordinates provide a unique way to express the location of a point in the plane. Instead of describing a point by its horizontal and vertical distances, like in the Cartesian coordinate system, polar coordinates use a radial distance and an angle.
  • The distance from the origin to the point is called the radial coordinate, denoted by \( r \).
  • The angle measured from the positive x-axis to the line connecting the point and the origin is the angular coordinate, denoted by \( \theta \).
This system is particularly useful when dealing with curves and shapes that are best described by rotations and distances, such as circles and spirals. By using polar coordinates, complex trigonometric relationships can often be handled more easily, especially in the context of polar graphs like the lima莽on.
Dimpled Lima莽on
The dimpled lima莽on is a fascinating type of curve. It is described in polar coordinates by the equation \( r = a + b \cos\theta \) or \( r = a + b \sin\theta \), where the constants \( a \) and \( b \) dictate the curve's shape.When \(|a| > |b|\), the shape becomes what we call a "dimpled" lima莽on. Here are some important features:
  • Shape: The curve resembles a lima莽on with a slight inward curve or "dimple" and lacks an inner loop.
  • Range: The values of \( r \) for a dimpled lima莽on range from \( a - |b| \) to \( a + |b| \), which means it never actually becomes zero.
Because the equation never equates to zero, a dimpled lima莽on does not pass through the origin. This property distinguishes it from other types of lima莽ons, like those with loops or cardioids. Understanding this distinction can help in graphing polar equations and predicting the behavior of polar curves.
Cardioid
A cardioid is a special type of lima莽on that has a heart-like shape, hence the name "cardioid." It is defined by the condition \(|a| = |b|\) and takes the form \( r = a + a \cos\theta \) or \( r = a + a \sin\theta \).Key characteristics of a cardioid include:
  • Geometry: The shape is symmetrical and looks like a rounded heart. It has just one "dimple" that reaches the origin, creating a single cusp or point.
  • Origin Passage: The cardioid passes through the origin exactly once where the equation \( r = 0 \) is satisfied.
  • Rotation: Depending on whether cosine or sine is used, the cardioid's orientation in the plane will change (e.g., left-right versus up-down orientation).
Cardioids appear in various mathematical contexts and are an excellent example of how polar equations translate into visual shapes. Recognizing when an equation forms a cardioid can help you better interpret and solve problems involving polar graphs.

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Most popular questions from this chapter

Find an equation for the parabola that satisfies the given conditions. (a) Focus (6,0)\(;\) directrix \(x=-6\) (b) Focus (1,1)\(;\) directrix \(y=-2\)

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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 $$d=\sqrt{r_{1}^{2}+r_{2}^{2}-2 r_{1} r_{2} \cos \left(\theta_{1}-\theta_{2}\right)}$$

Prove: The line tangent to the ellipse $$ \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 $$ at the point \(\left(x_{0}, y_{0}\right)\) has the equation $$ \frac{x x_{0}}{a^{2}}+\frac{y y_{0}}{b^{2}}=1 $$

Use the following values, where needed: $$\text { radius of the Earth }=4000 \mathrm{mi}=6440 \mathrm{km}$$ $$\text { 1 year (Earth year) =365 days (Earth days) }$$ $$1 \mathrm{AU}=92.9 \times 10^{6} \mathrm{mi}=150 \times 10^{6} \mathrm{km}$$ (a) Let \(a\) be the semimajor axis of a planet's orbit around the Sun, and let \(T\) be its period. Show that if \(T\) is measured in days and \(a\) is measured in kilometers, then \(T=\left(365 \times 10^{-9}\right)(a / 150)^{3 / 2}\). (b) Use the result in part (a) to find the period of the planet Mercury in days, given that its semimajor axis is \(a=57.95 \times 10^{6} \mathrm{km}\). (c) Choose a polar coordinate system with the Sun at the pole, and find an equation for the orbit of Mercury in that coordinate system given that the eccentricity of the orbit is \(e=0.206\) (d) Use a graphing utility to generate the orbit of Mercury from the equation obtained in part (c).
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