Problem 16 - Calculus Early Transcendentals

Question

(a) Find the eccentricity, (b) identify the conic, (c) give an equation of the directrix, and (d) sketch the conic. $$r=\frac{10}{5-6 \sin \theta}$$

Step-by-Step Solution

Answer

Eccentricity is 1.2, the conic is a hyperbola, directrix: \(y=8.33\).

Step 1: Identify the Form of the Equation

The equation given is \(r=\frac{10}{5-6\sin \theta}\). We can identify it in the form \(r=\frac{ed}{1+e\sin\theta}\). This is a polar equation of a conic section with its focus at the origin. Let's identify the eccentricity \(e\) and the directrix \(d\).

Step 2: Compare and Find Eccentricity (e)

By comparing the given equation \(r=\frac{10}{5-6\sin \theta}\) with the standard form \(r=\frac{ed}{1-e\sin\theta}\), we get that \(ed=10\) and \(-e=-6\sin\theta\). Thus, \(e=\frac{6}{5}=1.2\).

Step 3: Determine the Type of Conic

The eccentricity \(e=1.2\) is greater than 1. Therefore, the conic is a hyperbola.

Step 4: Equate and Solve for Directrix (d)

Still considering \(ed=10\), where \(e=1.2\), we solve for \(d\) by substituting \(e\) into the equation: \(1.2d=10\), which gives \(d=\frac{10}{1.2}\approx 8.33\). Therefore, the equation of the directrix is \(y=8.33\).

Step 5: Sketch the Conic

To sketch, we note that since the equation involves \(\sin \theta\), the directrix is horizontal. The focus is at the origin, and the curve opens vertically. Using the values of \(d\) and \(e\), sketch a hyperbola symmetrical about the vertical axis with the transverse axis on the positive and negative y-axis.

Key Concepts

Eccentricity

Eccentricity is a key parameter in the study of conic sections such as ellipses, parabolas, and hyperbolas. It determines the shape of the conic by measuring how much it deviates from being circular. The eccentricity, denoted as \(e\), can provide insightful information about the conic's structure.

• If \(e < 1\), the conic is an ellipse, which means it is elongated but closed.
• If \(e = 1\), the conic is a parabola, representing a perfect open curve.
• If \(e > 1\), the conic is a hyperbola, indicating two diverging, open curves.

In this exercise, comparing the given equation \(r=\frac{10}{5-6\sin \theta}\) with its standard form reveals an eccentricity \(e = 1.2\). Since \(1.2 > 1\), we classify it as a hyperbola. This tells us that our conic section consists of two separate branches, each extending infinitely.

Polar Coordinates

Polar coordinates are a two-dimensional coordinate system where each point on a plane is determined by a distance from a reference point and an angle from a reference direction. They are especially useful for dealing with curves that repeat radially, such as circles, spirals, and various types of conic sections.

The focus of the conic is often located at the pole (origin) in polar coordinates, making it easy to visualize and analyze.

For the given equation \(r=\frac{10}{5-6\sin \theta}\), \(r\) represents the distance from the origin to any point on the conic, and \(\theta\) is the angle from the positive x-axis. This conversion from Cartesian coordinates to polar form simplifies the representation of the conic, revealing the properties of hyperbolas when using polar coordinates effectively.

Directrix

The directrix of a conic section serves as a fixed reference line used to define and draw the curve. In conjunction with the focus, it helps describe the conic using its eccentricity:

The directrix is particularly significant in hyperbolas and ellipses, providing a way to compare the distances involved:

• For a hyperbola, the directrix and focus together define the set of points where the ratio of the distance to the focus over the distance to the directrix is equal to the eccentricity.

In our previous calculations, the directrix \(d\) is derived using the relationship \(ed=10\). Solving for \(d\) with the calculated \(e = 1.2\), we find that \(d \approx 8.33\). The equation of the directrix is a horizontal line since the original polar equation incorporates \(\sin \theta\), resulting in \(y = 8.33\). The conic will be oriented such that it broadly matches this setup.

Hyperbola

A hyperbola is a fascinating geometrical structure found in conic sections, recognizable by two distinct, oppositely opening curves (branches). It emerges when the slice plane cuts through both nappes of a conical surface at an angle, not parallel to the base.

Key characteristics of hyperbolas include:

• They are symmetric about their transverse axis, which is the line that passes through the foci and the center of the hyperbola.
• Each branch approaches asymptotes that guide the hyperbola but never meet the curve.
• The equation given in terms of polar coordinates, \(r=\frac{10}{5-6\sin \theta}\), solidifies its identity as a hyperbola, with its focus at the origin.
• In a hyperbola, the property \(e > 1\) ensures this open structure.

To sketch a hyperbola with the known directrix \(y = 8.33\) and eccentricity \(e = 1.2\), ensure symmetry about the y-axis, with branches extending outward, respecting the directrix’s horizontal alignment.