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. The conic is a hyperbola opening downward.

Step 1: Identify the Form of the Equation

The given polar equation is \( r = \frac{7}{2 - 5 \sin \theta} \). This equation fits the standard form of a conic in polar coordinates \( r = \frac{ed}{1 - e \sin \theta} \), where \( e \) is the eccentricity and \( d \) is a constant. In our equation, that implies \( ed = 7 \), \( e \times 1 = 5 \), and \( e = 5 \).

Step 2: Calculate the Eccentricity

From the standard form comparison, we identified that \( e = 5 \). This value is the eccentricity of the conic. In conic sections, if \( e > 1 \), the conic is a hyperbola. Hence, this conic is a hyperbola.

Step 3: Determine the Directrix

Since \( ed = 7 \) and \( e = 5 \), we can find \( d = \frac{7}{5} \). Thus, the conic equation is in the form where the directrix is at \( r = \frac{7}{5} \). The directrix helps in understanding the structure and orientation of the conic.

Step 4: Sketch the Conic and Label Vertices

To sketch the conic, identify that it's oriented in the direction dictated by \( \sin \theta \), meaning it opens downward since the numerator is 2. The vertex, where \( r \) reaches a maximum or minimum value, is when \( \theta = \pi/2 \). Calculate \( r \) at \( \theta = 0 \) and \( \theta = \pi \) to find the vertices: \( r(\theta = 0) = \frac{7}{2} \) and \( r(\theta = \pi) = -\frac{7}{12} \). Mark and draw the hyperbola using these points and the natural hyperbola shape.

Key Concepts

Conic Sections

Conic sections are curves obtained by slicing a three-dimensional cone with a plane. Depending on the angle and position of the slice, one can obtain different types of curves: circles, ellipses, parabolas, and hyperbolas. These sections have distinct characteristics and equation forms in both Cartesian and polar coordinates.

• A circle is a special type of ellipse where all points are equidistant from a center point.
• An ellipse looks like a stretched circle and has two focal points.
• A parabola is a curved shape with one focus and a directrix; it opens outward with no boundary on one side.
• A hyperbola consists of two separate curves, each resembling mirrored parabolas.

Understanding these properties helps in analyzing the curve's behavior and appearance. We frequently use eccentricity, denoted by the symbol \( e \), to distinguish these forms. Eccentricity provides a measure of how "stretched" or "deviant" from being circular a conic section is. For instance, circles have an eccentricity of 0, ellipses have \( 0 < e < 1 \), parabolas have \( e = 1 \), and hyperbolas have \( e > 1 \). In polar form, identifying these conics involves recognizing the standard forms of polar equations.

Polar Coordinates

Polar coordinates offer a unique way of describing the position of a point in a plane, using a distance from a reference point and an angle from a reference direction. This system is especially useful for conic sections since many conic equations simplify when converted from Cartesian to polar form.
The position of a point is defined by two values:

• \(r\): the radial distance from the origin (center of polar coordinates)
• \(\theta\): the angle measured counterclockwise from the positive x-axis

To describe conic sections in polar coordinates, equations are often presented as \( r = \frac{ed}{1 - e\cos\theta} \) or \( r = \frac{ed}{1 - e\sin\theta} \), where \( e \) is the eccentricity and \( d \) portrays a scaling factor.
In polar coordinates, the layout of axes changes our perspective on these shapes. Circles appear as a constant radius, and ellipses, parabolas, and hyperbolas take forms dictated by their equations, revealing their intrinsic properties such as their orientation or elongation. The polar coordinate system is intuitive for problems involving angles and distances from a fixed point, making it a go-to tool for several physics and engineering applications.

Hyperbola

A hyperbola is a fascinating and complex conic section characterized by its two disconnected curves, termed "branches." These branches are mirror images of each other and open in opposite directions, indicative of a hyperbola's eccentricity being greater than 1. This unique structure leads to fascinating properties.

• The equation typically implies separation by rectangular units or polar forms. In polar coordinates, a hyperbola can be represented by \( r = \frac{ed}{1 - e\sin\theta} \) or \( r = \frac{ed}{1 - e\cos\theta} \), which depicts its symmetrical nature and orientation.
• The center of the hyperbola is at the midpoint of the traversing line, called the transverse axis, which connects the closest points on each branch.
• The vertices of the hyperbola are the points of closest proximity between the branches, aligning with the transverse axis.
• Additionally, hyperbolas have asymptotes, which are lines that the branches approach but never reach. These asymptotes help define the boundary and shape direction of the branches.

Understanding hyperbolas includes recognizing them through their distinct asymptotic behavior and openness, which contrasts them sharply from other conic sections like parabolas or ellipses. Hyperbolas often appear in applications such as in navigation systems or analyzing the spread of radio waves.