Problem 17 - Algebra and Trigonometry

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 a vertex at the origin and a directrix at \( y = -4 \).

Step 1: Identify the Type of Conic

The general polar equation of a conic with the focus at the pole is given by \( r = \frac{ed}{1 - e \sin(\theta)} \) or \( r = \frac{ed}{1 - e\cos(\theta)} \), where \( e \) is the eccentricity and \( d \) is the distance from the pole to the directrix. In the provided equation, \( r = \frac{4}{1 - \sin(\theta)} \), we identify that the expression matches \( r = \frac{ed}{1 - e\sin(\theta)} \) with \( ed = 4 \) and \( e = 1 \) since there is no coefficient in front of \( \sin(\theta) \). This indicates that the conic is a parabola because \( e = 1 \).

Step 2: Determine the Vertex

For a parabola with \( e = 1 \), the directrix is \( d \) units from the focus. We have \( ed = 4 \) and \( e = 1 \), so \( d = 4 \). The vertex of the parabola in the polar form lies on the conic at the closest approach, which in this case is at the pole (origin) with \( \theta = 0 \) since \( r \) becomes undefined there, marking the vertex in the polar system.

Step 3: Determine the Directrix

Given \( d = 4 \), the directrix of the parabola is 4 units below the pole as the sine function implies a vertical shift. Therefore, the directrix is the horizontal line \( y = -4 \) in Cartesian coordinates.

Step 4: Sketch the Graph

To sketch the graph, start with marking the origin as the vertex. Then draw the directrix, which is the line \( y = -4 \). Since it is a parabola, plot several points by substituting various angles \( \theta \) (like \( \pi/6, \pi/4, \pi/3 \)) into the polar equation to find corresponding \( r \) values. Sketch a smooth curve through these points traditionally opening towards the pole and away from the directrix. Indicate the vertex and directrix on the graph.

Key Concepts

Conic Sections

Conic sections refer to the curves that one can obtain by intersecting a plane with a double-napped cone. This means you can form different shapes depending on how you slice through. The main types of conic sections are:

• Circles
• Ellipses
• Parabolas
• Hyperbolas

Each of these shapes has its own set of unique properties and equations. They are important in various fields of science and mathematics due to their intriguing properties and real-world applications. Polarity in equations defines how these sections emerge based on varying parameters.

Eccentricity

Eccentricity is a key concept in understanding conic sections. It describes how "stretched out" a conic is. The eccentricity (\(e\)) is a non-negative real number that uniquely characterizes each conic section:

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

In this exercise, an eccentricity of 1 indicates that the shape in question is a parabola, which is a unique kind of conic where the curve is equally distant from a central line, known as the directrix, as it is to a fixed point called the focus.

Directrix

The directrix in the context of conic sections serves as a reference line. When dealing with polar equations, understanding the location of the directrix assists in identifying the nature of the conic section. It acts as a guide, dictating how the conic is oriented:

• For parabolas, there is always one directrix.
• In polar form, it helps find the vertex and sketch the conic accurately.
• The relationship between the vertex, focus, and directrix defines the characteristics and orientation of the parabola in the graph.

As per our exercise, the directrix is a horizontal line situated at \(y = -4\) in Cartesian coordinates, indicating its role in constructing the parabola.

Parabola

Parabolas are one of the four main types of conic sections characterized by their U-shaped curve. They occur when the plane slicing through the cone is parallel to its side. Parabolas exhibit the following properties:

• The eccentricity (\(e = 1\)) differentiates them from other conic sections.
• The distance from the focus to any point on the curve is equal to the distance from that point to the directrix.
• In polar coordinates, it is defined by equations such as \(r = \frac{4}{1 - \sin \theta}\)\ which confirms its parabolic nature.

CWith an equation given in polar coordinates, finding the closest approach or the vertex gives us insight into the structure and position of the parabola.

Vertex

The vertex of a conic section, particularly a parabola, is its most central and significant point. Understanding its position sets the stage for defining the parabola:

• In a parabola, the vertex acts as the turning point or the peak.
• It is equally distant from the focus and directrix.
• In polar coordinates, for the equation \(r = \frac{4}{1 - \sin \theta}\)\, it is positioned at the pole, marking the closest approach of the parabola to the focus.

Identifying the vertex helps complete the graph and align the parabola accurately relative to its directrix and other coordinates.