Problem 46 - College Algebra

Question

For the following exercises, find the polar equation of the conic with focus at the origin and the given eccentricity and directrix. Directrix: \(y=-2 ; e=\frac{1}{2}\)

Step-by-Step Solution

Answer

The polar equation is \(r = \frac{2}{2 + \sin(\theta)}\).

Step 1: Understand the Relation Between Directrix, Eccentricity, and Polar Equation

A conic section with a focus at the origin and a given eccentricity \(e\) can be expressed in polar coordinates by the equation \(r = \frac{ed}{1 + e\sin(\theta)}\) or \(r = \frac{ed}{1 + e\cos(\theta)}\). To decide which form to use, we need to determine if the directrix is vertical (\(x = \text{constant}\)) or horizontal (\(y = \text{constant}\)).

Step 2: Identify the Directrix Orientation

The given directrix is \(y = -2\), which is a horizontal line. For horizontal directrices, use the sine function in the denominator: \(r = \frac{ed}{1 + e\sin(\theta)}\).

Step 3: Determine Values for Parameters

The eccentricity \(e\) is given as \(\frac{1}{2}\), and the directrix is \(y = -2\), so \(d = 2\) because the distance from the pole is the absolute value of the directrix constant.

Step 4: Formulate the Polar Equation

Substitute \(e = \frac{1}{2}\) and \(d = 2\) into the formula: \[ r = \frac{\frac{1}{2} \times 2}{1 + \frac{1}{2}\sin(\theta)} = \frac{1}{1 + \frac{1}{2}\sin(\theta)} \]. This simplifies to \[ r = \frac{2}{2 + \sin(\theta)} \].

Key Concepts

eccentricity

Eccentricity is a fundamental concept when dealing with conic sections. It's a measure that describes how much a conic deviates from being circular.

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

Eccentricity helps in identifying the shape and nature of the conic section. In the given problem, the eccentricity is \(\frac{1}{2}\), indicating the conic is an ellipse. The smaller the eccentricity, the closer the figure is to being circular. This property is crucial for understanding the geometrical structure and behavior of the conic in a plane.

directrix

The directrix plays a vital role in defining conic sections. It is a fixed line used to describe the set of points that form the conic section, in conjunction with the focus. In polar coordinates, the relationship between the directrix and focus helps in formulating the equation of the conic.

• The directrix can be either vertical or horizontal, affecting the equation form.
• A vertical directrix aligns with the cosine function.
• A horizontal directrix, like in this exercise where \(y = -2\), aligns with the sine function.

To utilize the directrix in the equation, it's important to take the absolute value of the directrix constant when calculating its distance from the origin. This process is vital in determining the polar equation of the conic.

polar coordinates

Polar coordinates are a coordinate system that uses the distance from a reference point (usually the origin) and an angle from a reference direction to specify points in a plane. This is different from the more common Cartesian coordinate system, which uses x and y values. In polar coordinates, a point is represented by \((r, \theta)\):

• \(r\) is the radius or distance from the origin.
• \(\theta\) is the angle from the reference direction, typically the positive x-axis, measured counterclockwise.

Polar equations, like \(r = \frac{2}{2 + \sin(\theta)}\) in the exercise, express relationships using \(r\) and \(\theta\). They are particularly useful for dealing with conic sections where the focus is at the origin, as they simplify the mathematics involved in describing curves that spiral outwards or remain circular.

conic sections

Conic sections are the curves obtained by slicing a cone with a plane. Depending on the angle and position of the slice, the resulting shape can be a circle, ellipse, parabola, or hyperbola. Here are some key aspects of conic sections:

• Ellipses and circles are closed curves.
• Parabolas and hyperbolas are open curves.
• Conic sections are defined by their eccentricity and focus-directrix relationship.

For conics situated in polar coordinates with the focus at the origin, equations are devised using the directrix and eccentricity to define them uniquely, as demonstrated in the exercise. This allows a straightforward derivation of their equations, such as using sine or cosine in the denominator depending on the directrix alignment. Conics are key in various fields, from astronomy to architecture, representing paths of celestial objects and design curves, respectively.