Problem 17 - 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. $$r=\frac{4}{1-\sin \theta}$$

Step-by-Step Solution

Answer

The conic is a parabola with vertex at (0,0) and directrix at y = -4.

Step 1: Identify the Type of Conic

The given polar equation can be recognized in the form \(r = \frac{ed}{1 + e\sin\theta}\) or \(r = \frac{ed}{1 - e\sin\theta}\). The conic is determined by the eccentricity \(e\). In the given equation, \(e = 1\) (by comparing with \(r = \frac{4}{1-\sin\theta}\)), which from the definitions of conics corresponds to a parabola.

Step 2: Sketch the Conic Graph

Since \(e=1\), the graph of the conic is a parabola. The factor \(4\) in \(\frac{4}{1-\sin\theta}\) will determine the size of the parabola, but not its shape. The parabola opens downwards because of the subtraction in the denominator. Sketch a parabola that opens downwards from the pole.

Step 3: Find the Vertex of the Parabola

In a polar equation of the form \(r = \frac{d}{1 - \sin\theta}\), the vertex is at the pole when \(\theta = 0\). So, the vertex for this parabola is at the origin \((0, 0)\) in the polar coordinate system.

Step 4: Identify the Directrix

For a polar parabola \(r = \frac{d}{1 - \sin\theta}\), the directrix is at \(y = -d\). Here, \(d=4\), so the directrix is the line \(y = -4\). Indicate this line on the graph as below the parabola.

Key Concepts

Parabola

A parabola is a unique type of conic section that displays a symmetrical curve. Unlike other conics such as ellipses and hyperbolas, a parabola only boasts a single curve or path. This special shape occurs when a plane intersects a cone parallel to its slope. In the case of the problem at hand, the equation \(r = \frac{4}{1 - \sin \theta}\) describes a parabola in a polar coordinate system.

In general, the parabolic shape is defined such that every point on it is equidistant from a specific point (the focus) not on the line known as the directrix. For the given polar equation, when we compare it to the standard form \(r = \frac{ed}{1 - e\sin\theta}\), it becomes evident that the eccentricity \(e = 1\), which distinctively characterizes a parabola.

Eccentricity

Eccentricity is a critical factor when identifying conic sections. It essentially describes how much a conic section deviates from being circular. For different values of eccentricity \(e\):

• 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.

In our specific problem, the equation \(r = \frac{4}{1 - \sin\theta}\) reveals the eccentricity \(e = 1\) when closely compared to its general form. This confirms for us that the conic section presented in the polar equation is indeed a parabola. This characteristic helps us instantly recognize the conic without needing to graph it.

Vertex and Directrix

The vertex and directrix are fundamental elements in identifying and sketching extended elements of a parabola. The vertex is the parabola’s "turning point," and it’s critical in defining the axis of symmetry for the conic.

For this particular equation, \(r = \frac{4}{1 - \sin\theta}\), the vertex can be located at the pole due to its alignment with the standard polar form. It results in the vertex occurring at the origin \((0, 0)\) when \(\theta = 0\).

The directrix, on the other hand, is a line opposite to the pole from the vertex. It is a reference line that dictates the reflective property of the parabola. By using the equation of our parabola, it's deduced that the directrix rests at \(y = -4\), marking it parallel to the x-axis below the vertex.

Polar Coordinate System

The polar coordinate system is an innovative way to define positions on a plane, using a radius and angle, rather than the usual \((x, y)\) coordinate pair in Cartesian form. In the polar system, a point is represented by \((r, \theta)\), where \(r\) is the radial distance from the origin (the pole), and \(\theta\) is the angle measured from the positive x-axis.

This type of system becomes particularly handy when dealing with conic sections like circles, ellipses, parabolas, and hyperbolas. Conics can be represented neatly in this system as specific equations based on their eccentricity and directrix relations.

For our parabola example \(r = \frac{4}{1 - \sin\theta}\), the pole, being the center, simplifies the identification of the vertex and allows straightforward plotting due to the angle-based measurements which indicate the parabola's orientation. Understanding this system enables a more intuitive grasp of the elegance of conics in mathematics.