Problem 33 - Thomas Calculus

Question

Exercises \(29-36\) give the eccentricities of conic sections with one focus at the origin, along with the directrix corresponding to that focus. Find a polar equation for each conic section. $$ e=1 / 2, \quad x=1 $$

Step-by-Step Solution

Answer

The polar equation of the conic section is \( r = \frac{1}{2 - \cos \theta} \).

Step 1: Understand the Relationship for a Conic Section in Polar Form

The polar equation of a conic section with one focus at the origin is given by \( r = \frac{ed}{1 + e \cos \theta} \) or \( r = \frac{ed}{1 - e \cos \theta} \), depending on the location of the directrix. In this case, since the directrix is \( x = 1 \), we will use \( r = \frac{ed}{1 - e \cos \theta} \) because the focus is at the origin.

Step 2: Identify e and the Directrix

Here, the eccentricity \( e = \frac{1}{2} \). The directrix is \( x = 1 \). This gives us \( d = 1 \), which is the distance from the focus to the directrix.

Step 3: Substitute Values into the Polar Equation

Substitute \( e = \frac{1}{2} \) and \( d = 1 \) into the polar equation \( r = \frac{ed}{1 - e \cos \theta} \). Thus, the equation becomes \[ r = \frac{\frac{1}{2} \cdot 1}{1 - \frac{1}{2} \cos \theta} \].

Step 4: Simplify the Equation

Simplify the expression \( \frac{\frac{1}{2}}{1 - \frac{1}{2} \cos \theta} \) to obtain \( r = \frac{1/2}{1 - \frac{1}{2} \cos \theta} \). Multiply numerator and denominator by 2 to simplify further, yielding \[ r = \frac{1}{2 - \cos \theta} \].

Key Concepts

Eccentricity in Conic Sections

Eccentricity is a key parameter in defining the shape of a conic section. It is denoted by the letter \( e \). Understanding eccentricity helps us categorize conics into different types: circles, ellipses, parabolas, and hyperbolas. Here's a quick breakdown of how eccentricity works:

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

In our exercise, \( e = \frac{1}{2} \), which tells us that the conic is an ellipse. Eccentricity describes how "stretched out" a conic section is. An ellipse with an eccentricity of \( \frac{1}{2} \) is less elongated than one with \( e = 0.9 \). This is why understanding \( e \) is crucial.

The Role of the Directrix

The directrix is a fixed line associated with conic sections. It serves a vital role in defining the curve along with the eccentricity. The relationship between the directrix and the focus determines the exact shape and orientation of the conic:

• The directrix helps provide a reference line for constructing the conic.
• It assists in deriving the polar equation of the conic section.
• The perpendicular distance from any point on the conic to the directrix is proportional to the distance from the point to the focus, dictated by the eccentricity.

In the given problem, the directrix is \( x = 1 \). This piece of information tells us where the line is positioned in relation to the focus located at the origin. Knowing the directrix allows us to construct the polar equation, as outlined in the step-by-step solution.

Focus at the Origin and Polar Equations

The focus is a point related to the conic section and is vital in its definition. Placing the focus at the origin simplifies the formulation of the polar equation. Here's why:

• In polar coordinates, having the focus at the origin allows us to use standard forms for the conic equation.
• The equation’s form indicates whether the directrix is horizontal or vertical, which influences the substitution in the polar equation form \( r = \frac{ed}{1 \pm e \cos \theta} \).

With the focus at the origin and given \( e = \frac{1}{2} \), we use the equation \( r = \frac{1/2}{1 - \frac{1}{2}\cos \theta} \) and simplify it to \( r = \frac{1}{2 - \cos \theta} \). This form directly illustrates the balance between the focus, directrix, and eccentricity to define the conic's shape in the polar plane.