Question

For the following exercises, sketch the graph of each conic. $$ r=\frac{3}{2+6 \sin \theta} $$

Short answer

The conic is a hyperbola, vertically oriented.

Step-by-step solution

Identify the Conic Form

The given equation is in polar form: \( r = \frac{3}{2 + 6 \sin \theta} \). This form is similar to \( r = \frac{ed}{1 + e \sin \theta} \), which is the general form of a conic section in polar coordinates. Here, \( ed = 3 \), \( e = 6 \), and \( d = \frac{1}{2} \). The standard form indicates the conic is a hyperbola.

Calculate the Eccentricity

Eccentricity (\( e \)) determines the type of conic. In the given equation, \( e = 6 \). For conics, \( e = 1 \) indicates a parabola, \( e < 1 \) indicates an ellipse, and \( e > 1 \) indicates a hyperbola. Since \( e = 6 > 1 \), it confirms the conic is a hyperbola.

Analyze the Conic's Orientation

The form \( 2 + 6 \sin \theta \) suggests that the directrix is vertical (parallel to the polar axis). This equation will have the hyperbola opening upwards or downwards, centered along the vertical axis.

Determine Key Characteristics

Using \( r = \frac{3}{2 + 6 \sin \theta} \), calculate intersections with axes to find vertices. Substitute \( \theta = \frac{\pi}{2} \) (to maximize \( \sin \theta \)): \( r = \frac{3}{8} \). Substitute \( \theta = \frac{3\pi}{2} \): \( r = -\frac{3}{4} \). These values can guide plotting.

Sketch the Conic

Using the characteristics derived (hyperbola, vertical orientation), sketch the graph. Mark the key points (\( r = \frac{3}{8} \) and \( r = -\frac{3}{4} \)) and show the hyperbola opening vertically with these constraints.

Key concepts

Polar Coordinates

Polar coordinates present a unique way to represent points on a plane using a circular grid. Unlike the regular Cartesian system that uses \(x\) and \(y\) coordinates, polar coordinates use \(r\) (radius) and \(\theta\) (angle). This system is particularly useful for problems involving conic sections, as it relates directly to rotations and distances in a radial manner.

The radius, \(r\), specifies how far a point is from the origin, while the angle, \(\theta\), indicates the direction from the origin (measured in radians). For example, a point with polar coordinates \(r = 3\) and \(\theta = \frac{\pi}{2}\) is located at a distance of 3 units upwards, as \(\theta = \frac{\pi}{2}\) points along the positive y-axis.

In our exercise, the equation \(r = \frac{3}{2 + 6 \sin \theta}\) represents a conic section in polar form. Thus, understanding polar coordinates is crucial for interpreting this conic and its characteristics.

Eccentricity of Conics

Eccentricity is a vital parameter that helps us identify the type of conic section we are dealing with. It is denoted by the variable \(e\) and represents how "stretched" a conic is.

Different values of eccentricity determine different kinds of conics:

• For a parabola, \(e = 1\).
• For an ellipse, \(e < 1\).
• For a hyperbola, \(e > 1\).

In our given exercise, we calculated that \(e = 6\), which is greater than 1. Therefore, this clearly indicates that the conic is a hyperbola.

Understanding eccentricity helps us not only determine the type of the conic section but also provides insight into its geometric properties, like the shape and orientation of its branches.

Hyperbola Graphing

Hyperbolas are fascinating and complex conic sections that appear when \(e > 1\). They consist of two distinct curves called branches, which are mirror images of each other. In polar coordinates, hyperbolas have specific intersection points and spread according to the values of \(r\) and \(\theta\).

For the equation \(r = \frac{3}{2 + 6 \sin \theta}\), the graph of this hyperbola requires an understanding of key features. It opens vertically due to the \(\sin \theta\) in the denominator.

When graphing the hyperbola, ensure to mark:

• Vertices: Using angles to find these, such as \(\theta = \frac{\pi}{2}\) and \(\theta = \frac{3\pi}{2}\), where \(r\) values were \(\frac{3}{8}\) and \(-\frac{3}{4}\).
• Asymptotes: Imaginary lines that determine the hyperbola's shape.

Accurate plotting of these coordinates helps form the hyperbola's branches, highlighting the hyperbolic nature of areas beyond the vertices.

Conic Orientation

Conic orientation refers to the direction in which a conic section's principal features, like axes or branches, are aligned. It's crucial for predicting the graph's shape and how it opens.

In the context of our hyperbola, with the term \(2 + 6 \sin \theta\) in the denominator, it suggests that the orientation relates to the polar axis. This forms a vertically oriented hyperbola, opening upward and downward along the vertical axis.

Knowing the orientation helps one identify:

• How the graph should extend on the plane.
• Where the branches of the conic will appear.
• Alignment relative to other graph elements.

This makes sketching the hyperbola more intuitive, aiding in accurately portraying the graph's characteristics in the coordinate system.