Problem 18 - Algebra and Trigonometry

Question

15–22 (a) Find the eccentricity and identify the conic. (b) Sketch the conic and label the vertices. $$r=\frac{10}{3-2 \sin \theta}$$

Step-by-Step Solution

Answer

The conic is a hyperbola with eccentricity 2.

Step 1: Identify the form of the conic equation

The given polar equation is \( r = \frac{10}{3 - 2 \sin \theta} \). This equation compares to the general form of a conic in polar coordinates, \( r = \frac{ed}{1 - e \sin \theta} \), which indicates it's a conic with respect to a focus at the pole.

Step 2: Determine the parameters 'e' and 'd'

For comparison, the given equation can be rewritten to match \( r = \frac{ed}{1 - e \sin \theta} \). Here, \( ed = 10 \) and \( 1 - e \sin \theta = 3 - 2 \sin \theta \). From matching coefficients, we have:1. \( ed = 10 \)2. \( e = \frac{2}{1} = 2 \)

Step 3: Identify the eccentricity

From the step above, we've determined that the eccentricity \( e = 2 \). The value of eccentricity reveals the type of conic. For conics:- If \( e = 1 \), it's a parabola.- If \( e < 1 \), it's an ellipse.- If \( e = 0 \), it's a circle.- If \( e > 1 \), it's a hyperbola.Since \( e = 2 \) which is greater than 1, the conic is a hyperbola.

Step 4: Identify the conic

The conic is identified as a hyperbola since the eccentricity \( e = 2 \) and is greater than 1.

Step 5: Sketch the conic

To sketch the hyperbola, we note that the directrix is vertical since the equation is in the form \( r = \frac{ed}{1 - e \sin \theta} \). The vertices are found from polar coordinates by considering the extreme values of \( r \).

Key Concepts

Eccentricity

In the realm of conic sections, eccentricity is a fundamental property that helps in identifying and distinguishing various conic shapes. It is denoted by the symbol \( e \). The eccentricity determines the shape of the conic section:

• If \( e = 0 \), the conic is a circle, which is the most "balanced" shape.
• If \( 0 < e < 1 \), the shape is an ellipse, which is somewhat stretched.
• If \( e = 1 \), it indicates a parabola, showing a specific open curve.
• If \( e > 1 \), the conic section is a hyperbola, characterized by two separate branches.

For the given polar equation \( r = \frac{10}{3 - 2 \sin \theta} \), the eccentricity is \( e = 2 \). This value, being greater than 1, confirms that the conic in question is a hyperbola.

Conic Sections

Conic sections are the curves obtained by slicing a double cone. These sections capture a variety of shapes:

• A circle, which is a curve where all points are equidistant from the center.
• An ellipse, which resembles a "squashed" circle, with two focal points.
• A parabola, which has a distinctive curved shape with one focal point.
• A hyperbola, consisting of two mirrored, open curves known as branches.

Each conic section can be described by its eccentricity. Circle and ellipse form closed shapes, whereas parabola and hyperbola do not. Understanding these sections is crucial because they model many natural and man-made structures.

Hyperbola

A hyperbola is a fascinating type of conic section that has unique properties compared to other conic sections. It consists of two disconnected curves called branches. These branches are symmetrical around a center point.

• The eccentricity \( e \) of a hyperbola is always greater than 1.
• In the context of the given polar equation \( r = \frac{10}{3 - 2 \sin \theta} \), the equation of a hyperbola indicates that as \( \theta \) changes, \( r \) changes, allowing the branches to extend indefinitely.
• Hyperbolas can often model systems involving orbits and escape velocities, making them crucial in celestial mechanics.

Despite their complex appearance, hyperbolas have straightforward mathematical descriptions, primarily determined by their eccentricity.

Polar Equation

Polar equations are a beautiful way to represent conics using the polar coordinate system. They relate a radial distance \( r \) and an angular coordinate \( \theta \).

• The general form for a conic in polar coordinates with a focus at the pole is \( r = \frac{ed}{1 - e \sin \theta} \) or \( r = \frac{ed}{1 - e \cos \theta} \).
• This polar form is particularly useful for identifying and describing conic sections when a focus is considered one of the poles.
• In our equation \( r = \frac{10}{3 - 2 \sin \theta} \), we derived \( e = 2 \) and thus identified it as a hyperbola. The structure provides insight into how \( r \) changes as \( \theta \) rotates.

Polar equations help map the points in their respective layouts, and illustrate the important features of the conic sections like directrices and focal points elegantly.

Vertex of Conic

The vertices of a conic are crucial points that help in sketching and understanding the overall shape of the conic section. In the case of a hyperbola, each branch has its own vertex.

• A vertex is the point on each branch where the conic section comes closest to or is furthest from its center.
• From the polar equation \( r = \frac{10}{3 - 2 \sin \theta} \), vertices can be identified by examining the extreme values of \( r \), which inform how far the curves extend.
• In practical terms, the vertices are essential for providing a framework to accurately sketch the shape of the hyperbola and understand its symmetry.

Vertices serve as essential reference points, anchoring the hyperbola’s expansive curves symmetrically around the center.