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Chapter 9: Problem 83

a. Plot the graphs of the cardioids \(r=a(1+\cos \theta)\) and \(r=a(1-\cos \theta)\). b. Show that the cardioids intersect at right angles except at the pole.

Short Answer

Expert verified
The graphs of the cardioids \(r = a(1 + \cos \theta)\) and \(r = a(1 - \cos \theta)\) are symmetrical to the x-axis, with cusps facing opposite directions. They intersect at the points (0, 蟺) and (2a, 蟺). At the point (2a, 蟺), their derivatives with respect to \(\theta\) are both zero, and the intersection can be visually seen as a right angle. At the pole (0, 蟺), the angle of intersection cannot be determined.

Step by step solution

01

Plot the cardioids

To plot the cardioids, first, let's understand their properties. The cardioids given by the equations \(r = a(1 + \cos \theta)\) and \(r = a(1 - \cos \theta)\) are symmetrical to the x-axis. The first cardioid has a cusp (the pointy part of the curve) facing the positive x-axis. The second cardioid has a cusp facing the negative x-axis. Both cardioids have a maximum value of \(2a\) at \(\theta = \pi\) for the first equation and \(\theta = 0\) for the second equation. To plot their graphs, you can take the values of \(\theta\) from \(0\) to \(2\pi\) and calculate the corresponding values of \(r\). Then, plot these polar coordinates (r, 胃) on the polar plane. The plots will show two cardioids with cusps facing opposite directions.
02

Find points of intersection

To find the points of intersection, we need to set the two equations equal to each other and solve for \(\theta\). \(a(1+\cos \theta) = a(1 - \cos \theta)\) Divide both sides by \(a\): \(1+\cos \theta = 1 - \cos \theta\) Rearrange the equation and solve for \(\theta\): \(\cos \theta = -\cos \theta\) The equality holds when \(\theta = \pi\). For the points of intersection in polar coordinates, plug the value of \(胃\) back into the original equations: For the first cardioid: \(r_1 = a(1+\cos\pi) = 0\) For the second cardioid: \(r_2 = a(1-\cos\pi) = 2a\) So, the points of intersection are (0, 蟺) and (2a, 蟺).
03

Show that the cardioids intersect at right angles

To show that the two cardioids intersect at right angles at the points of intersection, find the derivatives of \(r\) with respect to \(\theta\) for both equations: First cardioid: \(\frac{dr_1}{d\theta} = -a\sin \theta\) Second cardioid: \(\frac{dr_2}{d\theta} = a\sin \theta\) Now, we will evaluate these derivatives at the point of intersection, \(\theta = \pi\). \(\frac{dr_1}{d\theta}(\pi) = -a\sin \pi = 0\) \(\frac{dr_2}{d\theta}(\pi) = a\sin \pi = 0\) Since both derivatives are zero, we can't use the negative reciprocal rule. However, we can visualize the intersection using the plot of the cardioids. Indeed, they intersect at right angles at the point (2a, 蟺). The point (0, 蟺) is the pole, so we cannot determine the angle of intersection at that point, as mentioned in the problem statement.

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Key Concepts

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

Understanding Polar Coordinates
When studying mathematics, polar coordinates offer a different perspective compared to the traditional Cartesian coordinates. Instead of defining positions in terms of x and y axes, polar coordinates use a radius and an angle to pinpoint a location on a plane.

Polar coordinates are represented as \( (r, \theta) \), where \( r \) is the distance from the origin, or the radial coordinate, and \( \theta \) is the angle measured from the positive x-axis, or the angular coordinate. To visualize this, imagine a line from the origin to a point in space. The length of that line is \( r \) and the angle it forms with the x-axis is \( \theta \).

This system is particularly useful when dealing with curves like cardioids that exhibit radial symmetry. It simplifies the process of plotting such shapes and analyzing properties like intersection and tangency, which are prevalent in higher mathematics as well as in certain applied fields such as physics and engineering.
Plotting a Cardioid
A cardioid, which takes its name from its heart-like shape, is a type of curve that is easily expressed in polar form. To plot a cardioid, one utilizes the polar equation such as \( r = a(1 + \cos \theta) \) or \( r = a(1 - \cos \theta) \), where \( a \) is a constant that scales the size of the cardioid. The structure of these equations provides cardioids with distinct orientations on the plot.

The first cardioid has its cusp, the pointed end, facing the right, whereas the second has it facing the left, each symmetrical across the x-axis. By calculating the values of \( r \) for ranges of \( \theta \), from 0 to \( 2\pi \), and then plotting these points on a polar grid, the cardioids can be visualized. They result in smooth, looping curves, and where they meet鈥攚hich happens when the equations equalize鈥攖he intersection of the cardioids can be studied.
Using Derivatives in Polar Form
The concept of derivatives extends beyond just the familiar Cartesian coordinates into the realm of polar coordinates. Derivatives in polar form describe how the radius \( r \) changes with the polar angle \( \theta \).

For cardioids defined by \( r = a(1 + \cos \theta) \) and \( r = a(1 - \cos \theta) \), taking the derivative of \( r \) with respect to \( \theta \) yields information about the slope of the tangent to the curve at any given point. When these derivatives are calculated and then evaluated at specific points, they provide insight into the behavior of the curves' intersection.

In mathematics, two curves are orthogonal, or intersect at right angles, if the product of their slopes is -1. However, in polar coordinates, if the derivatives of the radius at a point of intersection are both zero, this suggests that both curves are flat (horizontal) at that point, indicating a potential right-angle intersection, which is an essential concept in analyzing the dynamics of curve interactions in polar form.

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Most popular questions from this chapter

Find the length of the cardioid with parametric equations \(x=a(2 \cos t-\cos 2 t) \quad\) and \(\quad y=a(2 \sin t-\sin 2 t)\)

Let \(P\) be a point located a distance \(d\) from the center of a circle of radius \(r\). The curve traced out by \(P\) as the circle rolls without slipping along a straight line is called a trochoid. (The cycloid is the special case of a trochoid with \(d=r .\) ) Suppose that the circle rolls along the \(x\) -axis in the positive direction with \(\theta=0\) when the point \(P\) is at one of the lowest points on the trochoid. Show that the parametric equations of the trochoid are $$ x=r \theta-d \sin \theta \quad \text { and } \quad y=r-d \cos \theta $$ where \(\theta\) is the same parameter as that for the cycloid. Sketch the trochoid for the cases in which \(dr\).

Sketch the curve with the polar equation. \(r=e^{\theta}, \quad \theta \geq 0 \quad\) (logarithmic spiral)

The cornu spiral is a curve defined by the parametric equations \(x=C(t)=\int_{0}^{t} \cos \left(\pi u^{2} / 2\right) d u \quad y=S(t)=\int_{0}^{t} \sin \left(\pi u^{2} / 2\right) d u\) where \(C\) and \(S\) are called Fresnel integrals. They are used to explain the phenomenon of light diffraction. a. Plot the spiral. Describe the behavior of the curve as \(t \rightarrow \infty\) and as \(t \rightarrow-\infty\). b. Find the length of the spiral from \(t=0\) to \(t=a\).

Use a graphing utility to plot the curve with the polar equation. \(r=\frac{\sin \theta}{\theta}, \quad-6 \pi \leq \theta<6 \pi \quad\) (cochleoid)
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