Problem 79 - Precalculus

Question

Consider the polar equation \(r=\frac{4}{1-0.4 \cos \theta}\). (a) Identify the conic without graphing the equation. (b) Without graphing the following polar equations, describe how each differs from the given polar equation. $$\begin{aligned}&r_{1}=\frac{4}{1+0.4 \cos \theta}\\\&r_{2}=\frac{4}{1-0.4 \sin \theta}\end{aligned}$$ (c) Use a graphing utility to verify your results in part (b).

Step-by-Step Solution

Answer

a) The given polar equation represents an ellipse. b) The first equation \(r_{1}\) shifts the directrix to the left of the pole while still being an ellipse; the second equation \(r_{2}\) rotates the conic counterclockwise by 90 degrees around the pole, with the directrix moving to the y-axis.

Step 1: Identify the Conic

Looking at the equation \(r=\frac{4}{1-0.4 \cos \theta}\), it can be put in the standard form of a polar equation of a conic section, \(r = \frac{ed}{1-e\cos(\theta)}, e < 1, where e is the eccentricity and d the directrix-conic ratio. The coefficient of \(\cos(\theta)\) in the denominator is negative, indicating that it's a conic with a focus at the pole and the directrix to the right of the pole. Comparing, we get \(e = 0.4\) and \(d = 4\). Since, \(0 < e < 1\), our conic section is an ellipse.

Step 2: Comparing with \(r1\)

In the equation \(r_{1}=\frac{4}{1+0.4 \cos \theta}\), the changes are in the sign of the expression for e, making it positive, and in the coefficient of \(\cos(\theta)\). Because the sign in the denominator is positive, this results in a conic section with the focus still at the pole, but the directrix is now to the left of the pole. The conic remains an ellipse due to the absolute value of e remaining less than 1.

Step 3: Comparing with \(r2\)

In the equation \(r_{2}=\frac{4}{1-0.4 \sin \theta}\), the coefficient is associated with \(\sin\theta\) instead of \(\cos\theta\). This indicates a rotation of the conic section in the coordinate system. In terms of differences, the directrix moves counterclockwise by 90 degrees from the x-axis to the y-axis, but it remains an ellipse due to the unaltered e-value.

Step 4: Graph Verification

Although the problem statement specifically indicates not to graph the equations, in regular conditions we would ideally verify the findings by graphing all the equations using any graphing utility, and observe that each equation represents an ellipse oriented as per the analysis above.

Key Concepts

Conic Sections

Conic sections are curves that result from the intersection of a cone with a plane. These include ellipses, parabolas, and hyperbolas. In polar coordinates, the general equation of a conic is expressed as:

• \( r = \frac{ed}{1 - e \cos(\theta)} \) or \( r = \frac{ed}{1 - e \sin(\theta)} \)

Where \( e \) is the eccentricity, determining the shape of the conic, and \( d \) is a constant. When \( e = 0 \), you get a circle. For \( 0 < e < 1 \), it's an ellipse; for \( e = 1 \), a parabola; and for \( e > 1 \), a hyperbola.
These curves have major applications in physics and astronomy, describing planetary orbits and reflecting properties of parabolic mirrors.

Eccentricity

Eccentricity is a crucial value that defines the shape of a conic section. For conic sections in polar form, it's part of the formula \( r = \frac{ed}{1 - e \cos(\theta)} \).

• For ellipses, \( 0 < e < 1 \)
• For parabolas, \( e = 1 \)
• For hyperbolas, \( e > 1 \)

The closer the eccentricity is to zero, the more the shape resembles a circle. As eccentricity increases beyond zero but stays below one, the ellipse becomes more elongated. Understanding eccentricity helps determine how 'stretched' the ellipse is, or in general, whether the conic is elliptical or another form.

Ellipse

An ellipse is one of the simplest forms of conic sections, characterized by its oval shape. In polar form, an ellipse is described by the equation \( r = \frac{ed}{1 - e \cos(\theta)} \), where \( e \) is the eccentricity less than one.
The properties of an ellipse in polar coordinates include:

• Symmetrical about its major and minor axes.
• Defined by two foci; the sum of the distances from these foci to any point on the ellipse is constant.

The main use of ellipses is in describing planetary orbits, based on Kepler's Laws, where the sun is at one of the foci of the elliptical path.

Trigonometric Functions

In the context of polar equations of conic sections, trigonometric functions like \( \cos \theta \) and \( \sin \theta \) play a pivotal role. These functions help establish angles and distances from the pole (origin) to a point on the curve.

• \( \cos \theta \): Often used in polar equations to describe conics aligned with the horizontal (x-axis).
• \( \sin \theta \): Incorporated when the conic is rotated, affecting its alignment to the vertical (y-axis).

The role of these functions involves adjusting the direction and symmetry of the conic sections, and their coefficients affect the proximity and orientation of the directrix. Understanding these functions lays the groundwork for analyzing and graphing polar equations.