Problem 18 - Precalculus: Mathematical for Calculus

Question

(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 an eccentricity of 2.

Step 1: Identify the Conic Type

The given polar equation is \( r = \frac{10}{3 - 2 \sin \theta} \). This equates to the standard form \( r = \frac{ed}{1 + e \sin \theta} \), indicating a conic section. Comparing with the form \( r = \frac{ed}{1 - e \sin \theta} \) we identify the conic’s general form. Here, the numerator is 10 and the denominator matches, therefore, \( e \sin \theta \) should have a negative sign switch, making \(-2\) becomes positive since negative appears upfront in standard equation. We deduce \(-2 = -e \), resulting in \( e = 2 \). Hence, numerator \(10\) matches \(2 \times d\), giving \( d = 5 \).

Step 2: Determine the Eccentricity

Using \( e = \frac{2}{1} = 2 \) fully aligns with identified equation terms. Because \( e > 1 \), this signifies a hyperbola. Hyperbolas always have eccentricities greater than 1.

Step 3: Sketch the Conic

In polar coordinates, observe placing center at usual origin. The hyperbola opens vertically due to \( \sin \theta \) influencing an upwards or downwards orientation. It is essential to focus on positive \( 2 \) multiplying \( \sin \theta \), which affects opening direction related to regular cosine or sine dominance over hyperbola stretch. Draw it opening towards maximum and minimum sine (+ and - values on unit circle points).

Step 4: Identify Vertices

Hyperbola vertices can be calculated from principal axis location. Generally, these define at closest and furthest based on interplay numerator 10/2 offset. Here, they depend deeply on encapsulations cyclical generated pattern emanates 5-threatening specific sine maximum points, aligning along \( y\) polar positive-negative curve maxima.

Key Concepts

Eccentricity

Eccentricity is a crucial concept in understanding conic sections. It essentially measures the deviation of a conic from being circular. The value of eccentricity determines the type of conic:

• If the eccentricity (\( e \)) is less than 1, the conic is an ellipse.
• If the eccentricity is equal to 1, the conic is a parabola.
• If the eccentricity is greater than 1, as is the case in our example, the conic is a hyperbola.

In this particular problem, we found the eccentricity to be 2, which clearly indicates that the conic is a hyperbola. This value suggests that the curve extends infinitely in opposite directions, emphasizing the open nature of a hyperbola. Understanding eccentricity helps in visualizing the shape and extent of the conic.

Polar Coordinates

Polar coordinates offer a unique way of mapping points in a plane using a radius and an angle. Unlike the Cartesian system, which uses \( x \) and \( y \) coordinates, polar coordinates define positions using \( r \) (the distance from the origin) and \( \theta \) (the angle from the positive x-axis). This system is particularly helpful when dealing with curves like circles and conics.In polar coordinates, conics are typically represented in the form:\[ r = \frac{ed}{1 \, \pm \, e \sin \theta} \hspace{5px} \text{or} \hspace{5px} r = \frac{ed}{1 \, \pm \, e \cos \theta} \]These representations allow us to understand how the radius changes as the angle varies, which provides insights into the conic’s orientation and opening. For example, in the given equation \( r=\frac{10}{3 - 2 \sin \theta} \), the factor affecting the sinusoidal nature in the denominator influences the shape of the hyperbola.

Hyperbola

A hyperbola is one of the four types of conic sections and is characterized by its distinctive open shape. It consists of two separate, mirror-image branches which extend infinitely. Each branch of a hyperbola is symmetrically distributed about the transverse axis. Key features of a hyperbola include:

• Vertices: The vertices are points where each branch is closest to the other. They lie on the transverse axis.
• Foci: Beyond the vertices are the foci, the points from which distances are measured to generate the hyperbola.
• Asymptotes: These are the lines which the hyperbola approaches but never touches. They guide the shape of the hyperbola.

In polar equations, as in the provided exercise, the denominator often indicates the interaction of sine or cosine functions. Sine and cosine can determine how the hyperbola is oriented. An equation involving sine, like in this exercise, will result in a vertical orientation of the hyperbola openings. The hyperbola will stretch vertically, following the maximum and minimum points of sine on the polar grid.