91Ó°ÊÓ

Eccentricity is a key parameter in defining the shape of a conic section. It is denoted by the letter \( e \). For different values of \( e \), we have different conic sections:

• 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 given problem, the eccentricity is found to be 2, meaning \( e = 2 \). Because \( e > 1 \), we know the conic section in our exercise is a hyperbola.

A hyperbola is one of the four conic sections and is defined by its eccentricity \( e \), which is greater than 1. A hyperbola consists of two distinct branches that open away from each other. The standard forms of hyperbolas are:

• \( xy \text{-plane: } \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \)
• \( polar coordinates: } r = \frac{ed}{1 + e \text{ (some trigonometric function)} } \)

The given equation of our hyperbola is \( r = \frac{1}{1 - 2 \boldsymbol{\text{sin}} \theta } \). By comparing it to the standard polar form \( r = \frac{ed}{1 + e \boldsymbol{\text{sin}} \theta } \), we confirm \( e = 2 \). The hyperbola will have its focus at the pole and opens away from the directrix line.

In conic sections, the directrix is a fixed line used in conjunction with the focus to define a conic. For a hyperbola, the directrix helps to establish the set of points that form the conic. The equation for the directrix in polar coordinates, given the eccentricity \( e \) and the parameter \( d \), is:
\[ r = \frac{ed}{1 + e \boldsymbol{\text{sin}} \theta} \]
For the problem, since \( e = 2 \) and by setting \( d = 1 \), the equation of our directrix simplifies to:
\[ r = \frac{2 \times 1}{1 + 2 \boldsymbol{\text{sin}} \theta} = 1 \]
This line will be horizontal and at \( r = 1 \), and it guides the shape and orientation of the hyperbola.

Polar coordinates represent a point in the plane by a distance and an angle rather than by Cartesian \((x, y)\) coordinates. The distance \( r \) is from the origin, and the angle \( \theta \) is measured counterclockwise from the positive x-axis. In terms of conic sections, polar coordinates allow us to write complex curves more simply. For instance, the general form of a conic in polar coordinates is:
\[ r = \frac{ed}{1 + e \text{ (some trigonometric function)} \theta} \]
In our exercise, we used this form to express a hyperbola as \( r = \frac{1}{1 - 2 \boldsymbol{\text{sin}} \theta} \). This form clearly shows the relationship between the radius \( r \), the angle \( \theta \), and the eccentricity \( e \). Working in polar coordinates can simplify the understanding and visualization of conic sections.