Problem 35 - Precalculus Mathematics for Calculus

Question

(a) Find the eccentricity and identify the conic. (b) Sketch the conic and label the vertices. $$r=\frac{7}{2-5 \sin \theta}$$

Step-by-Step Solution

Answer

Eccentricity is 5 and the conic is a hyperbola.

Step 1: Identify the Conic Equation

The given polar equation is in the form \( r = \frac{ed}{1 + e \sin \theta} \). However, observe that it matches the other form involving sine: \( r = \frac{ed}{1 - e \sin \theta} \). This confirms we have a conic with a focus at the origin and a directrix perpendicular to the polar axis.

Step 2: Determine the Eccentricity

In the standard form \( r = \frac{ed}{1 - e \sin \theta} \), the eccentricity \( e \) and equation \( r = \frac{7}{2 - 5 \sin \theta} \) we have \( ed = 7 \) and \( e = 5 \). Therefore, \( d = \frac{7}{5} \) and \( e = 5 \).

Step 3: Identify the Conic Type

The value of eccentricity \( e = 5 \) is greater than 1, which indicates that the conic is a hyperbola. For ellipses, \( e < 1 \); for parabolas, \( e = 1 \); and for hyperbolas, \( e > 1 \).

Step 4: Sketch the Conic

To sketch the hyperbola, note that the conic opens in the direction where \( \theta = 270^{\circ} \), downwards from the pole, because of the negative sign in \( 2 - 5 \sin \theta \). It crosses the polar axis and moves away from it drastically.

Step 5: Label the Vertices

The vertices are located along the line perpendicular to the polar axis. Use the information \( r = 0 \) to find the intersection points and consider further calculations outside polar, for approximate graphical representation, confirm maximum extension.

Key Concepts

Conic Sections

Conic sections are fascinating curves that result from the intersection of a plane with a double-napped cone. They are classic subjects in mathematics due to their geometric and algebraic properties. The main types of conic sections are:

• Circle
• Ellipse
• Parabola
• Hyperbola

Each type of conic is defined based on its eccentricity, denoted by the symbol \( e \):

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

Eccentricity helps to understand the 'shape' or the 'deviation from being circular.' Hyperbolas, as mentioned in the original solution, have an eccentricity greater than 1 which makes them exhibit two diverging branches.
These sections are prevalent not only in geometry but also in physics, engineering, and astronomy, where orbits of planets and satellites are typically elliptical or hyperbolic.

Polar Coordinates

Polar coordinates provide a unique way of describing a point in the plane based on its distance from the origin and the angle with respect to the horizontal axis called the polar axis. It is an alternative to Cartesian coordinates:

• Radius \( r \): The distance from the origin (pole) to the point.
• Angle \( \theta \): The angle formed with the positive x-axis measured in radians or degrees.

Polar equations generally of the form \( r = f(\theta) \) allow describing curves like circles, spirals, and conics. The transition from Cartesian to polar coordinates is particularly useful for studying symmetrical properties and solving problems involving curves that are circular or spiral in nature.
In conic sections, the polar equation is often used to represent them with a focus at the origin. The equation from the exercise \(r=\frac{7}{2-5 \sin \theta}\) is an example of a polar equation representing a hyperbola.

Hyperbola

A hyperbola is a type of conic section characterized by its open, curved paths called branches, which look like two mirrored, infinite arcs. Hyperbolas are defined by their eccentricity, \( e \), greater than 1. This is the main indicator that differentiates it from other conics like ellipses or parabolas.
The hyperbola's defining equation in polar coordinates can be written in the standard form \( r = \frac{ed}{1 - e \sin \theta} \) or \( r = \frac{ed}{1 - e \cos \theta} \). In our exercise, the equation \( r = \frac{7}{2 - 5 \sin \theta} \) fits this form, confirming its nature as a hyperbola.When graphing a hyperbola using polar coordinates:

• The direction in which the hyperbola opens can be inferred from the sign in the denominator (\(-e \sin \theta\) or \(-e \cos \theta\)).
• The vertices of the hyperbola lie along the principal axis, which can be determined based on the angle \( \theta \).

Understanding hyperbolas in polar coordinates is essential in various fields including astronomy, where celestial objects sometimes follow hyperbolic trajectories, especially when influenced by gravity and other forces.