Problem 3 - Precalculus: Mathematical for Calculus

Question

Write a polar equation of a conic that has its focus at the origin and satisfies the given conditions. Parabola, directrix \(y=2\)

Step-by-Step Solution

Answer

The polar equation is \( r = \frac{2}{1 + \sin \theta} \).

Step 1: Identify the conic type

The given conic is a parabola, which indicates that the eccentricity, \( e \), is equal to 1. This is because for parabolas, \( e = 1 \).

Step 2: Determine the position of the directrix

The directrix is given as \( y = 2 \). In polar coordinates, a horizontal line above the origin can be represented using the equation \( y = -p \) for parabolas. Since \( y = 2 \), we identify \( -p = 2 \). Thus, \( p = -2 \). This means the directrix is 2 units above the pole (origin).

Step 3: Write the polar equation for the parabola

The general polar equation for a parabola with a horizontal directrix is \( r = \frac{ed}{1 + e \sin \theta} \). Here, \( e = 1 \) and \( d \) is the distance from the focus to the directrix, which in this case \( d = 2 \) since the line is 2 units away from the origin. Therefore, the equation becomes \( r = \frac{2}{1 + \sin \theta} \).

Key Concepts

Conic Sections

Conic sections are shapes created as a plane intersects a double-napped cone. There are four main types: circles, ellipses, parabolas, and hyperbolas. These shapes have been studied since ancient times and are essential in various fields such as astronomy, engineering, and physics.
Understanding conic sections helps in visualizing geometric relationships and solving complex problems. Each conic section can be expressed using specific equations, and they are characterized by their eccentricity, denoted by \( e \):

• **Circle**: \( e = 0 \), a special form of an ellipse.
• **Ellipse**: \( 0 < e < 1 \), an elongated circle.
• **Parabola**: \( e = 1 \), a symmetrical open curve used in satellite dishes and headlights.
• **Hyperbola**: \( e > 1 \), curves forming two disjointed arcs.

Conic sections have diverse applications, and their properties help to understand the pathway of planets and the design of optical lenses. The specific focus here is on parabolas, where the conic intersects the cone in a way that forms a mirror-symmetrical curve.

Parabola

A parabola is one of the four primary types of conic sections. It is defined as the set of all points equidistant from both a fixed point called the "focus" and a line known as the "directrix." This definition geometrically illustrates why the shape is ideal for reflecting light and sound in applications like parabolic antennas and mirrors.
In polar coordinates, parabolas are characterized by their eccentricity \( e = 1 \). This means that the distance from any point on the curve to its focus is equal to the distance to its directrix. The polar equation for a parabola with a horizontal directrix is represented as:
\[ r = \frac{ed}{1 + e \sin \theta} \]
where \( r \) is the distance from the origin to a point on the parabola, \( d \) is the perpendicular distance from the focus to the directrix, and \( \theta \) is the angle from the reference direction to the line drawn from the pole to any point on the parabola.
The beauty of parabolas lies in their reflective properties; rays of light that enter parallel to the axis of symmetry converge at the focus. This intrinsic feature is why parabolas are prevalent in various engineering solutions.

Directrix

The directrix of a conic section is a fundamental element that, together with the focus, helps in defining and constructing shapes like parabolas. Directrices can be seen as the geometric mirror line for each point on the parabola, maintaining consistent distances with the focus.
For a parabola defined in polar coordinates, the directrix is crucial in forming the parabola itself. It is usually a straight line, and any point on the parabola is equidistant from the directrix and the focus. For instance, if the polar equation involves a vertical directrix, the equation might take the form:
\[ r = \frac{ed}{1 + e \sin \theta} \]
This directrix aids in understanding the parabola's orientation. When the directrix is given as \( y = c \), like \( y = 2 \) in the original exercise, it hints at how the parabola will open and where it will be located around the focus.
Understanding the concept of a directrix helps visualize and analyze the structure of a parabola, making it easier to comprehend the spatial relationships and predict the shape's behavior in practical applications. This interplay is fundamental in grasping the dynamics of trajectories and the spread of forces and energies in physics and engineering.