Problem 18 - Precalculus: Mathematics for Calculus

Question

A polar equation of a conic is given. (a) Show that the conic is a parabola, and sketch its graph. (b) Find the vertex and directrix, and indicate them on the graph. $$r=\frac{3}{2+2 \sin \theta}$$

Step-by-Step Solution

Answer

The conic is a parabola with its vertex at the origin and the directrix at y = -3. It opens downwards.

Step 1: Identify the conic form

The given polar equation is \( r = \frac{3}{2 + 2\sin\theta} \). This matches the general form for conics \( r = \frac{ed}{1 + e\sin\theta} \) with \( e = 1 \) and \( d = 3 \). The eccentricity \( e = 1 \) indicates that the conic is a parabola.

Step 2: Vertex and directrix

For the equation \( r = \frac{ed}{1 + e\sin\theta} \) with \( e = 1 \), \( d = 3 \), and in this form \( r = \frac{3}{1 + \sin\theta} \), the vertex of the parabola is at \( \theta = 270^\circ \) or \( 3\pi/2 \) (since the form \( r = \frac{ed}{1 + e\sin\theta} \) aligns the vertex at the direction of the sine term). The directrix is perpendicular to the polar axis and at a distance \( d = 3 \) units from the pole, along the negative y-axis (since \( +\sin \) corresponds to \( y \)).

Step 3: Sketch the graph

To sketch the parabola, note that it opens toward the negative y-axis because the equation in its current form directs this behavior. Place the vertex at the origin in polar coordinates and draw the parabola opening downward. The directrix is drawn as a horizontal line below the origin at distance 3 units.

Key Concepts

Parabola

A parabola is a unique type of conic section that can be described as the set of all points in a plane equidistant from a fixed point called the focus and a fixed line known as the directrix. In polar coordinates, a parabola's equation can vary, but one common representation is

• \( r = \frac{ed}{1 + e\sin \theta} \) or \( r = \frac{ed}{1 + e\cos \theta} \)

where \( e \) is the eccentricity and \( d \) is a constant. If the parabola is described in terms of sine, it typically opens downward or upward, depending on the sign of the sine component. Given the typical properties of a parabola, understanding how it derives from a circle or ellipse in conic sections is essential. Parabolas do not have a center like circles or ellipses and have a distinct shape characterized by their symmetrical open curve. The focus of a parabola significantly influences its shape and size.In our specific exercise, the given equation \( r = \frac{3}{2+2\sin \theta} \) is shown to be a parabola with eccentricity \( e = 1 \), indicating a unique geometric property. The value of \( e = 1 \) is fundamental in identifying it as a parabola because it signifies that the distances from any point on the curve to the fixed point, and fixed line, are equal.

Eccentricity

Eccentricity is a crucial concept in the study of conic sections, as it defines the shape of the curve. The eccentricity, denoted as \( e \), is a non-negative number that differentiates the type of conic:

• If \( e = 0 \), the conic is a circle.
• If \( 0 < e < 1 \), the conic is an ellipse.
• If \( e = 1 \), the conic is a parabola.
• If \( e > 1 \), the conic is a hyperbola.

Eccentricity determines how "stretched" or "elongated" a curve is. For parabolas, the value \( e = 1 \) implies that they neither stretch into a complete circle nor narrow into an ellipse or hyperbola. Instead, they maintain consistent divergence through their vertex. This openness is a crucial marker of a parabola's properties in conic sections.Understanding eccentricity helps in sketching and predicting the behavior of the conic sections. It plays a vital role when working with polar coordinates, where the eccentricity not only defines the type of the conic section but also influences its attributes such as the location of the vertex and the direction of the conic's opening.

Conic Sections

Conic sections are the curves obtained from the intersection of a plane with a double-napped cone. There are four primary types of conic sections: circles, ellipses, parabolas, and hyperbolas, each defined by its eccentricity. These curves have various equations in Cartesian and polar coordinates, which help us describe their geometric properties.In polar coordinates, conic sections are often written in the form

• \( r = \frac{ed}{1 + e\sin \theta} \)
• \( r = \frac{ed}{1 + e\cos \theta} \)

where the type of conic is dependent on the eccentricity \( e \). For the given problem, the polar equation \( r = \frac{3}{2+2\sin \theta} \) describes a parabola since the eccentricity \( e = 1 \).Each type of conic has its unique properties and applications. Circles and ellipses are often used in design and architecture, while parabolas find relevance in physics, particularly in projectile motion and satellite dishes due to their reflective properties. Hyperbolas also appear in various scientific applications including radio communications. By analyzing the polar forms, one can efficiently solve problems related to geometry and real-world applications.