Problem 9 - College Algebra

Question

Identify the conic with a focus at the origin, and then give the directrix and eccentricity. $$ r=\frac{5}{1+2 \sin \theta} $$

Step-by-Step Solution

Answer

The conic is a hyperbola with eccentricity 2 and a directrix at \( d = \frac{5}{2} \).

Step 1: Identify the Conic Type

The given equation is of the form \( r = \frac{ed}{1 + e \sin \theta} \). This indicates that the conic section is represented in polar coordinates. In this standard polar equation, if the form is \( 1 + e \sin \theta \), then the conic section has its focus at the origin and the directrix is parallel to the polar axis.

Step 2: Determine the Eccentricity

In the given equation \( r = \frac{5}{1 + 2 \sin \theta} \), compare it to the standard form \( r = \frac{ed}{1 + e \sin \theta} \) to identify the eccentricity \( e \). We conclude that \( e = 2 \). The value of the eccentricity \( e \) determines the type of conic: if \( e = 2 \), which is greater than 1, the conic is a hyperbola.

Step 3: Find the Directrix

The directrix \( d \) can be found using the equation \( ed = 5 \). Substituting \( e = 2 \), we have \( 2d = 5 \). Solving for \( d \), we get \( d = \frac{5}{2} \). Therefore, the directrix is a line located at \( \frac{5}{2} \) units away from the pole, positioned such that the directrix is parallel to the line \( \theta = \frac{\pi}{2} \).

Key Concepts

Polar Coordinates

Polar coordinates provide an alternate way to describe the location of points in a plane. Instead of using the typical Cartesian system (x, y), we use

• the radial distance from the origin \( r \), and
• the angle \( \theta \) from the positive x-axis.

This system is particularly handy when dealing with curves like circles, ellipses, and other conic sections. For instance, a point's location in polar coordinates is represented as \( (r, \theta) \).
For conic sections, we often see equations that use these polar coordinates, as shown in the original exercise. The equation \( r = \frac{5}{1 + 2 \sin \theta} \) not only gives us the shape's relationship with its focus but also tells us about its alignment and orientation in polar coordinates. With the focus at the origin, this description becomes concise and meaningful, making it easier to analyze the conic's properties in this frame.

Eccentricity

Eccentricity is a key number that helps in classifying conic sections. It denotes how "stretched" a conic section is.

• If the eccentricity \( e \) is less than 1, the figure is an ellipse.
• If \( e \) equals 1, it forms a parabola.
• When \( e \) is greater than 1, as in our example where \( e = 2 \), the conic is a hyperbola.

This important property isn't just a number; it decides the overall shape and breadth of the curve. In the polar equation \( r = \frac{ed}{1 + e \sin \theta} \), the value of \( e \) works with \( \theta \) to dictate how the curve opens and its spread. In hyperbolas, the higher the eccentricity, the wider the branches of the hyperbola are.

Hyperbola

A hyperbola consists of two parts, known as branches, and these branches open either horizontally or vertically. When a hyperbola is defined in polar coordinates, as in our exercise, its equation particularly reveals its dynamic drawing.
Since the eccentricity \( e > 1 \), the given equation \( r = \frac{5}{1 + 2 \sin \theta} \) shows a hyperbola.

• The directrix for this hyperbola is found parallel to one of the coordinate axes, maintaining a fixed distance from the focus.
• In this case, the directrix is positioned at \( \frac{5}{2} \) units away, parallel to the line \( \theta = \frac{\pi}{2} \).

Hyperbolas are fascinating, as they are unique among conic sections for having two disjoint components and being defined by an asymptotic behavior. Their presence in various scientific applications, including understanding orbits and acoustics, underscores their importance.