Problem 15 - Precalculus

Question

Identify the conic and sketch its graph. $$r=\frac{3}{1-\cos \theta}$$

Step-by-Step Solution

Answer

Given the polar equation \(r=\frac{3}{1-\cos \theta}\), the conic section is a parabola that opens to the left.

Step 1: Transformation of the Polar Equation

First, transform the equation to a recognizable form of a conic section. The general form of a conics equation in polar coordinates is \(r=\frac{ed}{1±e\cos(\theta-\theta_0)}\), where e is the eccentricity, d is the distance from the pole to the directrix, and ± depends on whether the conic is opening to the right (positive) or left (negative).Substituting \(3\) for \(ed\) and \(1\) for \(e\), we have \(r=\frac{3}{1-\cos \theta}\), which matches the equation given. A cosine with a negative inside the parenthesis generally indicates that the conic will open to the left or right.

Step 2: Identifying the Conic Section

We can identify the conic section by looking at the eccentricity \(e\). For \(e<1\), the conic is an ellipse, for \(e=1\) it's a parabola, and for \(e>1\) it's a hyperbola. Because here \(e=1\), the conic section is a parabola.

Step 3: Graphing the Conic Section

The graph of a polar equation is plotted on a polar coordinate system, where \(r\) is the distance from the origin (pole) and \(\theta\) is the angle made with the positive x-axis. As our equation does not contain a \(\theta\) term, our parabola will open to the left (negative). From this, start by plotting points for varying \(\theta\) values, and noting the corresponding \(r\) values. The origin of our polar coordinate system is the focus of the parabola. Sketch the graph accordingly. Remember we only consider positive \(r\) values.

Key Concepts

Conic Sections

Conic sections are the curves obtained by slicing a double cone with a plane. Depending on the angle and position of the cut, different shapes emerge, each with unique features and equations. These shapes include:

• Ellipses: Oval shapes where the sum of the distances from two fixed points (foci) to any point on the ellipse is constant. If the eccentricity is less than 1, the conic is an ellipse.
• Parabolas: These look like U-shaped curves and occur when the plane is parallel to the cone's side. They occur when the eccentricity equals 1, as seen in the example equation, \( r=\frac{3}{1-\cos(\theta)} \).
• Hyperbolas: Characterized by two separate curves that mirror each other. A hyperbola appears when the plane cuts through both cones completely. This occurs with eccentricities greater than 1.

Understanding these foundational shapes helps in identifying and graphing conic sections, especially in polar coordinates.

Eccentricity

Eccentricity is a critical concept when examining conic sections. It determines the shape of the conic section based on its value. This concept can be represented in the conic's general equation in polar form, \( r = \frac{ed}{1±e\cos{(\theta-\theta_0)}} \).
Here are the types of conic sections based on eccentricity, \( e \):

• If \( e = 0 \), the figure is a circle.
• If \( 0 < e < 1 \), an ellipse is formed, which appears when the eccentricity is less than one.
• If \( e = 1 \), it transforms into a parabola, like in our example \( r=\frac{3}{1-\cos(\theta)} \).
• If \( e > 1 \), it results in a hyperbola.

Each conic section's shape adapts to the eccentricity, revealing the nature of the curve and how stretched or compressed it is compared to a circle. Understanding eccentricity allows us to decode the geometric properties and identity of these curves.

Graphing Polar Coordinates

Graphing polar coordinates involves plotting points on a coordinate plane that uses a polar grid. Here, each point is determined by a radius \( r \) and an angle \( \theta \), measured from the positive x-axis.
To graph a polar equation like \( r=\frac{3}{1-\cos \theta} \), follow these steps:

• Identify the conic's properties from its equation, such as eccentricity and orientation.
• Compute values of \( r \) for various \( \theta \) angles to create a set of points.
• Plot these points on the polar grid, noting that \( \theta \) is the angle from the positive x-axis and \( r \) is the distance from the pole (origin).
• Note that we only consider positive \( r \) values for drawing the graph.
• Connect the points to form the shape of the conic, ensuring accuracy in terms of orientation and structure based on the original equation parameters.

This process helps visualize conic sections and other curves on polar grids, providing insight into their structures and symmetries.