Problem 66 - Precalculus

Question

Determine whether the statement is true or false. Justify your answer. The graph of \(r=4 /(-3-3 \sin \theta)\) has a horizontal directrix above the pole.

Step-by-Step Solution

Answer

The statement is False. The graph of the polar equation \(r=4 /(-3-3 \sin \theta)\) has a horizontal directrix but it is below the pole, rather than above it.

Step 1: Understand the Structure of the Polar Equation

The equation given is a polar equation of a hyperbola in the format \(r = ed / (e - cos(\theta))\), where e is the eccentricity and d is the distance from the focus to the directrix. The denominator e - cos(\theta) or e - sin(\theta) tells us whether directrix lies above or below the pole. If it's minus cos, the directrix is vertical and if it's minus sin, the directrix is horizontal.

Step 2: Analyze the Given Equation

In the given equation \(r=4 /(-3-3 \sin \theta)\), the eccentricity e is 3 and distance d is 4. The sign in front of sin function is minus. This means the directrix is horizontal. If the denominator is negative, i.e., -3, it indicates that the directrix is below the pole.

Step 3: Conclusion

Analyzing the format, the sign and the function used in the denominator of the given equation, it's determined that the directrix is horizontal but it's below the pole, not above the pole.

Key Concepts

Hyperbolas

A hyperbola is a type of conic section that forms an open, two-branched curve. Unlike other conic sections such as ellipses and circles, hyperbolas have a distinguishing feature where each branch appears in separate parts of the plane.
In a polar coordinate system, hyperbolas can be described using polar equations. These equations typically involve the sine or cosine of the angle \(\theta\), which defines the position of any point along the curve relative to the pole.
What makes hyperbolas unique in polar equations is their eccentricity, which is always greater than one. This attribute creates the wide-open branches typically associated with hyperbolas. Understanding the structure of these equations is key to identifying the position and direction of the branches in relation to other elements, like the directrix.

Eccentricity

Eccentricity, represented by the letter \(e\), is a measure of how "stretched" a conic section is. For hyperbolas, \(e > 1\), which means they appear more "open" than ellipses or circles. In a polar equation, eccentricity influences the shape by determining how much the curve diverges from being a circle.
For a polar equation of the form \( r=\frac{ed}{e - \,\text{trig}\, \theta}\), the eccentricity \(e\) plays a crucial role in finding out the nature of the conic section. A larger \(e\) would mean the hyperbola's branches are further apart.
This parameter helps to understand the geometric properties of hyperbolas and is essential for determining other components such as the position of the directrix.

Directrix

The directrix is a fixed line used in the definition and description of the properties of conic sections. In polar equations representing hyperbolas, the directrix helps describe the orientation of the hyperbola in relation to the pole (the origin of the polar coordinate system).
In the context of the equation \(r=\frac{ed}{e - \sin \theta}\), the nature of the trigonometric function (either sine or cosine) helps to determine whether the directrix is vertical or horizontal. Specifically, \(-\sin \theta\) indicates a horizontal directrix.
The directrix can either be above or below the pole depending on the signs in the equation. If the denominator is negative overall, as it was in the exercise example, it means the directrix is below the pole.

Pole

The pole in polar coordinates is analogous to the origin in Cartesian coordinates. It's the point from which all distances in polar graphs are measured. In the polar plane, all points are described by the distance from the pole and the angle \(\theta\) from the positive horizontal axis.
Understanding the pole's role is crucial when working with polar equations like hyperbolas, where the distance from this point can determine where the directrix is positioned. In our exercise, the position of the directrix below the pole was determined based on the characteristics of the polar equation.
This concept helps students comprehend how various elements of the equation, such as eccentricity and trigonometric signs, converge around the pole to define the graph's properties.