Problem 17 - Calculus Early Transcendentals

Question

(a) Find the eccentricity and directrix of the conic \(r=1 /(1-2 \sin \theta)\) and graph the conic and its directrix. (b) If this conic is rotated counterclockwise about the origin through an angle 3\(\pi / 4\) . write the resulting equation and graph its curve.

Step-by-Step Solution

Answer

The eccentricity is 2; directrix is \(r=1\). The rotated equation is \(r = \frac{1}{1 + \sqrt{2}\cos\theta + \sqrt{2}\sin\theta}\).

Step 1: Identify the type of conic section

The given polar equation is \(r = \frac{1}{1 - 2\sin\theta}\). Here, \(e = 2\) (from the coefficient of \(\sin\theta\)). Since \(e > 1\), this is a hyperbola.

Step 2: Calculate the eccentricity

The eccentricity \(e\) of the conic is the coefficient of \(\sin\theta\) in the denominator, which is 2.

Step 3: Determine the directrix

The standard form for a conic in polar coordinates is \(r = \frac{ed}{1-e\sin\theta}\), where \(d\) is the distance from the pole to the directrix. Since \(d = 1\) for the given equation, the directrix is \(r = 1\).

Step 4: Graph the original conic and directrix

Graph the hyperbola in polar coordinates using \(r = \frac{1}{1 - 2\sin\theta}\). The directrix is a line at \(r = 1\). This graph shows a hyperbola with eccentricity 2 and a directrix parallel to the major axis.

Step 5: Rotate the conic equation

The given conic needs to be rotated counterclockwise by \(\frac{3\pi}{4}\) (135 degrees). To rotate the equation, use substitution for \(\theta\). Let \(\theta' = \theta - \frac{3\pi}{4}\) to get the rotated equation. Substituting this into the original, you get: \( r = \frac{1}{1 - 2\sin(\theta - \frac{3\pi}{4})}\). This simplifies to: \(r = \frac{1}{1 + \sqrt{2}\cos\theta + \sqrt{2}\sin\theta}\).

Step 6: Graph the rotated conic

Graph the rotated hyperbola using the equation \(r = \frac{1}{1 + \sqrt{2}\cos\theta + \sqrt{2}\sin\theta}\). This shows the hyperbola rotated by 135 degrees.

Key Concepts

Eccentricity

The eccentricity of a conic section is a key parameter that dictates its shape. It's denoted by the letter \(e\). For different values of eccentricity, we get different types of conic sections:

• If \(e = 0\), it's a circle.
• If \(0 < e < 1\), it's an ellipse.
• If \(e = 1\), it's a parabola.
• If \(e > 1\), it's a hyperbola.

Eccentricity shows how much a conic section deviates from being circular. In the given exercise, the equation \(r = \frac{1}{1 - 2\sin\theta}\) has an eccentricity of 2. This is because the equation is in the form where the coefficient of \(\sin\theta\) is the eccentricity. Since 2 is greater than 1, the conic is a hyperbola. Understanding eccentricity helps you identify and differentiate between conic sections at a glance.

Directrix

The directrix is a straight line used to define and construct a conic section. Each type of conic has its own relationship with this line:

• For parabolas, it is the line from which every point on the parabola is equidistant to the focus.
• For ellipses and hyperbolas, it's part of the geometric property that defines them, based on the ratio of distances from any point on the conic to the focus and to the directrix.

In polar coordinates, the equation \(r = \frac{ed}{1 - e\sin\theta}\) includes the directrix. Here, \(d\) is the distance from the origin. In our equation, \(r = \frac{1}{1 - 2\sin\theta}\), the directrix can be determined by getting \(d = 1\). Thus, the line \(r = 1\) serves as the directrix. Knowing the directrix helps visualize the orientation and sketch the conic accurately.

Polar Coordinates

Polar coordinates provide a unique way to express points on a plane. Instead of using \(x\) and \(y\) coordinates, polar coordinates use \(r\) and \(\theta\), where:

• \(r\) is the distance from a point to the origin.
• \(\theta\) is the angle formed with the positive x-axis.

This system is especially helpful when studying curves like conics, as it can simplify the equations involved. The exercise features a polar equation, \(r = \frac{1}{1 - 2\sin\theta}\), to describe a hyperbola. This format makes it easier to visualize rotations and transformations of the conic, as seen when we rotate the hyperbola by substituting \(\theta' = \theta - \frac{3\pi}{4}\). Understanding polar coordinates helps in analyzing curves that exhibit rotational symmetry or periodicity.

Hyperbola

A hyperbola is a type of conic section characterized by two distinct open curves. It appears when you slice through a double cone.

• Each point on the hyperbola maintains a constant difference in distance from two fixed points called foci.
• The hyperbola consists of two disconnected parts known as branches.

In the context of polar coordinates, hyperbolas have equations like \(r = \frac{1}{1 - e\sin\theta}\). This allows for easy identification of hyperbolas by checking if the eccentricity is greater than 1.Our original conic, \(r = \frac{1}{1 - 2\sin\theta}\), is a hyperbola with \(e = 2\), showing its characteristic open and diverging shape. Rotating the hyperbola produces a new orientation but keeps the essential diverging traits intact. Understanding hyperbolas aid in applications across physics and engineering, especially involving forces and paths represented by such curves.