Problem 68 - Precalculus with Limits

Question

TRUE OR FALSE? In Exercises 67-70, determine whether the statement is true or false. Justify your answer. The graph of \(r=\dfrac{4}{-3-3\ \sin\ \theta}\) has a horizontal directrix above the pole.

Step-by-Step Solution

Answer

False

Step 1: Identify the type of conic section

Examine the given polar equation. This equation \(r=\dfrac{4}{-3-3\ \sin\ \theta}\) takes the form \(r=\dfrac{p}{1+e\ \sin\ \theta}\) or \(r=\dfrac{p}{1-e\ \sin\ \theta}\). These are the standard forms of a polar equation of a conic section with the pole at one of the foci. Here, p=4 and e=-3. Since \(e<1\), this conic section will be a ellipse.

Step 2: Identify the directrix

For a conic section represented in polar coordinates, the directrix relates to the value of e and the denominator of the given equation. Since our e value is -3<0, this suggests that this ellipse will have a horizontal directrix below the pole (considering the polar coordinate system).

Step 3: Final evaluation

The original statement claims that there is a horizontal directrix above the pole. Given our analysis, the ellipse actually has a horizontal directrix below the pole. Therefore, the statement is false.

Key Concepts

Graph of Polar Equations

Polar equations graphically represent different types of conic sections like circles, ellipses, parabolas, and hyperbolas. These equations use polar coordinates, where each point on the plane is determined by a distance from the origin (called the 'pole') and an angle from a fixed direction.
One common form of polar equations for conic sections is \(r = \frac{p}{1 + e \sin \theta}\) or \(r = \frac{p}{1 - e \cos \theta}\), where:

• \(r\) is the distance from the pole,
• \(p\) is a parameter related to the size of the conic,
• \(e\) is the eccentricity, which defines the shape of the conic,
• \(\theta\) is the angle formed with respect to the positive x-axis.

The varying placement of the trigonometric functions **sine** or **cosine** determines orientation, specifically whether the directrix or symmetry axis is horizontal or vertical. Understanding these parameters helps in visualizing the orientation and shape of the conic in the polar coordinate system.

Ellipse in Polar Coordinates

An ellipse in polar coordinates has a distinct equation, usually represented as \(r = \frac{p}{1 + e \sin \theta}\) or \(r = \frac{p}{1 - e \cos \theta}\). For an ellipse:

• The eccentricity \(e\) is less than 1 \((e < 1)\), indicating that the conic does indeed form an ellipse, not a circle or other shape.
• The parameter \(p\) helps indicate the size and compactness of the ellipse.

In the specific problem discussed, the equation was \(r = \frac{4}{-3 - 3 \sin \theta}\), where \(p = 4\) and \(e = -3\). These values help determine properties of this ellipse like its orientation and size.
Regardless of eccentricity or the focal parameter \(p\), the ellipse's center remains at the focus or pole in a typical representation. Depending on the values \(e\), you find the elongation and flattening details of the ellipse.

Directrix in Polar Coordinates

The directrix in polar coordinates is a line related to the conic section's eccentricity and standard form. For the polar equation \(r = \frac{p}{1 + e \sin \theta}\) or \(r = \frac{p}{1 - e \cos \theta}\), the placement of the directrix depends on the parameter \(e\). Here's how:

• A positive \(e\) value indicates a vertical orientation of directrix, and a negative \(e\) typically points towards a horizontal orientation.
• When dealing with an equation like \(r = \frac{4}{-3 - 3 \sin \theta}\), the negative \(e\) suggests that the directrix is horizontal relative to the polar axis.

Importantly, for the equation in question, the directrix is below the pole due to this negative eccentricity. This affects how we understand the placement and symmetry of the conic within the polar system, influencing calculations regarding intersections or viewing angles.