Problem 21 - Algebra and Trigonometry

Question

15–22 (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

The conic is a hyperbola with eccentricity 5.

Step 1: Identify the Conic Section Form

The given equation is \( r = \frac{7}{2 - 5 \sin \theta} \). This is in the form \( r = \frac{ed}{1 + e \sin \theta} \), \( r = \frac{ed}{1 - e \sin \theta} \), \( r = \frac{ed}{1 + e \cos \theta} \), or \( r = \frac{ed}{1 - e \cos \theta} \). Since our equation is \( \frac{c}{d - e\sin\theta} \), it matches the form \( \frac{ed}{1 - e \sin \theta} \), indicating a vertical conic section centered at the pole (focus at the pole).

Step 2: Determine the Eccentricity

Compare the given equation \( r = \frac{7}{2 - 5 \sin \theta} \) to the standard form \( r = \frac{ed}{1 - e\sin\theta} \). Here, \( ed = 7 \) and \( e = 5 \), indicating \( e \) is the eccentricity. The eccentricity is 5, which categorizes the conic as a hyperbola, since \( e > 1 \).

Step 3: Identify the Type of Conic

Since the eccentricity \( e = 5 \), and as a property of conics, when \( e > 1 \), the conic section is a hyperbola.

Step 4: Sketch the Conic Section

To sketch a hyperbola given in polar coordinates, note the following: the hyperbola is symmetric with respect to the horizontal axis (since the sine term in the denominator involves \( \sin \theta \)). The center or pole (one of the foci) is at the origin. Plot the point where \( \theta = 0 \) (calculate \( r \) using \( \theta = 0 \)). The hyperbola opens vertically because of the sine term, extending towards the angles where \( \sin \theta \) approaches 1 (\( 90^\circ \) and 270°). Label the vertices located at these plotted points.

Key Concepts

Conic Sections

Conic sections are the curves obtained by intersecting a plane with a double-napped cone. This intersection can result in different types of curves depending on the angle and position of the intersecting plane relative to the cone.

• Circle: occurs when the intersection is perpendicular to the axis of the cone.
• Ellipse: similar to a circle, but stretched along one axis; it appears when the plane cuts through the cone at an angle, but does not go parallel to the cone's side.
• Parabola: occurs when the plane is parallel to one of the generatrices of the cone.
• Hyperbola: formed when the plane cuts through both nappes of the cone, at a steep angle.

Specifically, in polar coordinates, these conic sections can be represented as equations that describe their unique properties and shapes. It’s crucial to understand these forms as they provide information about the orientation and features of the conic section. In this exercise, you identified a hyperbola, which implies a certain level of steepness in the plane cutting through the cone.

Eccentricity

Eccentricity is a number that uniquely characterizes each conic section. It provides a measure of how much the conic section deviates from being circular. The value of eccentricity (\(e\)) determines the type of conic:

• Circle: \(e = 0\)
• Ellipse: \(0 < e < 1\)
• Parabola: \(e = 1\)
• Hyperbola: \(e > 1\)

For the hyperbola identified in the exercise, the eccentricity \(e\) was found to be 5. This high value signifies a hyperbola, which is a section of a cone where the plane intersects both nappes. Such a large eccentricity means the hyperbola branches are very spread out, with each branch extending off to infinity. This concept of eccentricity is what allows us to classify the type of conic section based solely on this value alone.

Hyperbola

A hyperbola is a fascinating type of conic section that is defined as the set of all points where the difference of the distances to two fixed points, known as foci, is constant. Unlike ellipses, which are loops, hyperbolas have open curves that go on indefinitely.
For a hyperbola given in polar coordinates as in the exercise, this is confirmed by the equation's form and the derived eccentricity \(e = 5\).
Key features of hyperbolas include two separate branches and asymptotes, which are lines that the curve approaches as it extends to infinity. In rectangular coordinates, hyperbolas have equations like \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\). However, in polar coordinates, they assume a different form and reveal the nature of focus points which are crucial for plotting.
Understanding hyperbolas in polar coordinates also implies recognizing how they are centered at the pole and how they reflect symmetry across specific axes. The inclusion of a sine term in our equation's denominator indicates that the hyperbola extends along the direction influenced by \(\theta\), helping in sketching the curve accurately. This mathematical beauty allows hyperbolas to model real-world phenomena such as orbit paths and signal propagation.