Problem 18 - Calculus Early Transcendentals

Question

Graph the conic \(r=4 /(5+6 \cos \theta)\) and its directrix. Also graph the conic obtained by rotating this curve about the origin through an angle \(\pi / 3\) .

Step-by-Step Solution

Answer

The conic is a hyperbola, rotated by \(\pi/3\) about the origin.

Step 1: Identify the Conic Type

The given equation of the conic is in the form \(r = \frac{ed}{1 + e \cos \theta}\), indicating a conic section in polar coordinates. Here, the parameters are \(e = 6\) and \(d = \frac{4}{5}\). Because \(e = 6 > 1\), this equation represents a hyperbola.

Step 2: Find the Directrix

The directrix of a conic in polar form of \(r = \frac{ed}{1 + e \cos \theta}\) is given by \(x = \frac{d}{e} = \frac{4/5}{6} = \frac{4}{30} = \frac{2}{15}\). Hence, the directrix is the vertical line \(x = \frac{2}{15}\).

Step 3: Initial Graph Construction

To graph the conic section, plot the polar equation \(r = \frac{4}{5 + 6 \cos \theta}\) over the range \(\theta = [0, 2\pi]\). Since it is a hyperbola, the graph will appear as two branches extending to infinity, and it should converge around the line \(x = \frac{2}{15}\).

Step 4: Rotation Transformation

To rotate the conic by \(\pi/3\), apply the transformation \(\theta = \theta' - \pi/3\). Substitute to get the new conic equation in terms of \(\theta'\): \[r = \frac{4}{5 + 6 \cos(\theta' - \pi/3)}\]Graph this equation by adjusting the angles for every point on the original conic.

Step 5: Graph the Rotated Conic

Using the transformed equation, again plot the graph over \(\theta' = [0, 2\pi]\). The entire conic, along with its directrix, should now be rotated counterclockwise by \(\pi/3\) radians (60 degrees) about the origin.

Key Concepts

Conic Sections

Conic sections are curves obtained by intersecting a plane with a double-napped cone. They are fundamental in the study of mathematics, particularly geometry. There are four primary types of conic sections: circle, ellipse, parabola, and hyperbola.
Certain properties help define and differentiate these shapes. For example, conic sections can be described both in Cartesian coordinates and polar coordinates. Polar coordinates use the variables \(r\) and \(\theta\) to represent points in a plane.

• Circle: Obtained when the cutting plane is parallel to the base of the cone.
• Ellipse: Occurs when the intersecting plane angles more steeply than that parallel to the base, but not enough to pass through both nappes.
• Parabola: Formed when the plane is parallel to a generator of the cone.
• Hyperbola: Produced when the plane intersects both nappes of the cone, as seen in our given polar equation.

Understanding these curves' properties and equations allows us to graph them and study their behaviors in various coordinate systems.

Hyperbola

A hyperbola is a type of conic section with an eccentricity \(e\) larger than 1. In simpler terms, the eccentricity is a measure of how much the conic deviates from being circular. For hyperbolas, this means their curves are more stretched than circles or ellipses.
In polar coordinates, a hyperbola can be expressed with the equation \(r = \frac{ed}{1 + e \cos \theta}\), where \(d\) is related to the distance from the directrix and \(e\) is the eccentricity.

• The hyperbola consists of two disconnected curves or branches.
• The branches extend infinitely and are mirror images of each other with respect to a central axis.
• The two branches converge or appear to meet at imaginary lines called asymptotes.

In this problem, our conic is represented by \(r = \frac{4}{5 + 6 \cos \theta}\). With \(e = 6\), it clearly indicates a hyperbola, given that 6 is greater than 1.

Directrix

The directrix of a conic section is a significant geometric concept. It is a fixed line used in the definition and formulation of the properties of conics. For a hyperbola in polar form, the directrix plays a crucial role in determining the curve's structure.
The formula for a directrix in polar coordinates is \(x = \frac{d}{e}\), where \(d\) is a scalar derived from the conic's equation, and \(e\) is the eccentricity.

• In the given exercise, \(d = \frac{4}{5}\), and \(e = 6\).
• Thus, the directrix is calculated as \(x = \frac{4/5}{6} = \frac{4}{30} = \frac{2}{15}\). This implies the hyperbola has a vertical line \(x = \frac{2}{15}\) serving as its directrix.

Having a directrix helps in graphically representing the flat side of the hyperbola and aids in understanding its complete structure in the plane.

Rotation Transformation

Rotation transformation in mathematics involves changing the position of a shape or figure in a plane around a fixed point, usually the origin. This technique is handy for visualizing conic sections in different orientations.
When we apply a rotation, each point of the shape moves along a circular path centered at the origin. In polar coordinates, we adjust by transforming the angle \(\theta\).

• To rotate a conic section by an angle \(\alpha\), replace \(\theta\) with \(\theta' - \alpha\) in its equation.
• In this exercise, a rotation of \(\pi/3\) is applied, meaning \(\alpha = \pi/3\).
• Therefore, the equation becomes \(r = \frac{4}{5 + 6 \cos(\theta' - \pi/3)}\).

By plotting this new equation, we can visualize how the hyperbola and its directrix have been rotated counterclockwise by 60 degrees. This transformation helps to analyze conics in varying orientations efficiently.