Question

For the following exercises, sketch the graph of each conic. $$ r=\frac{32}{3+5 \sin \theta} $$

Short answer

The polar equation represents a hyperbola with eccentricity \( e = \frac{5}{3} \).

Step-by-step solution

Identify the Polar Equation

The given equation is in polar form, specifically a rational equation involving a trigonometric function in the denominator: \( r = \frac{32}{3+5\sin\theta} \). It represents a conic section because it fits the general form for conics in polar coordinates: \( r = \frac{ed}{1 + e\sin\theta} \) or \( r = \frac{ed}{1 + e\cos\theta} \). Here, the form resembles \( r = \frac{ed}{1 - e\sin\theta} \) where we have \( ed = 32 \), and the multiple of \( \sin\theta \) and \( \cos\theta \) gives the eccentricity.

Determine the Conic Type

The equation \( r = \frac{32}{3+5\sin\theta} \) is in the form \( r = \frac{ed}{1 + e\sin\theta} \). We identify \( e = 5 / 3 \) by comparing it to the standard form \( r = \frac{ed}{1 + e\sin\theta} \). The eccentricity \( e > 1 \) indicates the conic is a hyperbola because it means the numerator is larger than the constant term in the denominator, aligning with the properties of hyperbolas.

Calculate the Directrix and Focus

For the polar equation \( r = \frac{ed}{1+e\sin\theta} \), the constant \( ed = 32 \). We use this relation to verify the components. Here, the directrix is typically a line parallel to the polar axis, and the focus is at the origin. Since \( e > 1 \), the conic's vertices lie along the line determined by \( \theta \) in the direction the asymptotes open.

Sketch the Graph

To sketch the graph of this hyperbola, note that for \( \theta = \frac{\pi}{2} \), \( r = \frac{32}{3+5} = 4 \), and for \( \theta = \frac{3\pi}{2} \), \( r = \frac{32}{3-5} \) suggests asymptotic behavior, thus the conic opens in the directions where the denominator approaches zero. The graph features branches that move away from the origin along the asymptotic lines. Plot key points and draw the conic open away from the directrix derived from the angles.

Key concepts

Polar Coordinates

Polar coordinates offer a different approach to representing points on a plane compared to the usual Cartesian coordinates. Instead of using x and y to describe positions, polar coordinates use two values: the radial distance from a central point (often called the pole or origin) and the angle from a reference direction (usually the positive x-axis).

• The radial coordinate, denoted as \( r \), measures how far the point is from the pole.
• The angular coordinate, denoted as \( \theta \), shows the direction of the point in relation to a fixed line, like the x-axis.

The polar equation given in the exercise, \( r = \frac{32}{3+5\sin\theta} \), helps to describe the position of points that form a conic shape. In this case, these coordinates facilitate understanding complex shapes by simplifying the mathematical expressions using angles and distances.

Eccentricity

Eccentricity is a fundamental concept for deciding the type of conic section a curve represents. It is a number that describes the shape of a conic, indicating how much it deviates from being a circle.

• If the eccentricity \( e = 0 \), the conic is a circle.
• If \( 0 < e < 1 \), it forms an ellipse.
• If \( e = 1 \), it represents a parabola.
• If \( e > 1 \), the shape is a hyperbola.

In our exercise, the polar equation is \( r = \frac{32}{3+5\sin\theta} \), and the eccentricity is determined as \( e = \frac{5}{3} \). Since \( e > 1 \), this indicates that the conic is a hyperbola. Eccentricity thus acts as a classification tool that tells us more about the shape by looking at its distance and opening.

Hyperbola

A hyperbola is a type of conic section that is generally formed when a plane intersects both nappes of a double cone. It is characterized by two flaring branches that spread out in opposite directions.Key features of a hyperbola include:

• It has two foci, which are the points around which the hyperbola is shaped.
• The transverse axis, which is the line segment that passes through the vertices of the hyperbola.
• Asymptotes, which are lines that the curve approaches but never touches. They guide the overall shape of the hyperbola.

In the given equation \( r = \frac{32}{3+5\sin\theta} \), the hyperbolic pattern is identified by the fact that the eccentricity is greater than one. The way the branches open and the nature of the asymptotes are often analyzed using its polar form, aiding in tracing the graph effectively.

Trigonometric Functions

Trigonometric functions like sine and cosine play an essential role in working with polar coordinates and conic sections. These functions help express the angular dependence of various shapes and positions.

• \( \sin(\theta) \) and \( \cos(\theta) \) relate to angles and can define segments of circles.
• They are periodic, meaning they repeat their values in regular intervals, known for their smooth oscillating behavior.

For the expression \( r = \frac{32}{3+5\sin\theta} \) seen in the exercise, \( \sin\theta \) helps determine changes in the distance \( r \) caused by variations in the angle \( \theta \). Trigonometric functions play a fundamental role in modifying conic equations, allowing them to model complex curves.