Problem 5 - Algebra and Trigonometry

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: Understanding Parabolas in Polar Coordinates

A parabola is a type of conic section where the eccentricity, denoted by \( e \), equals 1. This implies the conic is equidistant from the directrix and focus.

Step 2: Identify Relationship with the Directrix

Since the directrix is parallel to the x-axis at \( y = 2 \), it lies above the focus. The polar form of the equation of conics is \( r = \frac{ed}{1 - e \sin \theta} \). In this scenario, \( e = 1 \) and the directrix distance \( d = 2 \). Hence, we substitute these values into the equation.

Step 3: Substitute Values and Simplify

Substitute \( e = 1 \) and \( d = 2 \) into the polar equation format for conics, which gives: \[ r = \frac{2}{1 - \sin \theta} \]. This simplifies to the equation of the parabola.

Key Concepts

Conic Sections

Conic sections are the curves formed by the intersection of a plane and a double-napped cone. There are four primary types of conic sections: circles, ellipses, parabolas, and hyperbolas. Each type is determined by the angle at which the plane intersects the cone. - **Circle**: Forms when the plane cuts the cone parallel to the base. - **Ellipse**: Occurs when the plane cuts through the cone at an angle, but doesn't go through the base. - **Parabola**: Happens when the plane is parallel to the slope of the cone. - **Hyperbola**: Arises when the plane interacts with both nappes of the cone, but is not parallel to the axis. Each conic has unique properties that can be described using equations in both Cartesian and polar coordinates. Specifically for parabolas in polar equations, the focus is at the origin, and the structure of the parabola is largely determined by its directrix and eccentricity. Understanding conic sections helps in diverse fields like astronomy, architecture, and physics.

Eccentricity

Eccentricity is a key element in defining the shape and nature of a conic section. Denoted by the symbol \( e \), it describes how much conic sections deviate from being circular. - **Eccentricity values**: - **Circle**: \( e = 0 \) - **Ellipse**: \( 0 < e < 1 \) - **Parabola**: \( e = 1 \) - **Hyperbola**: \( e > 1 \)In a parabola, since \( e = 1 \), it indicates that each point on the parabola is equidistant from both the focus (a fixed point) and the directrix (a fixed line). This unique property creates the symmetrical shape characteristic of parabolas. For this specific exercise, understanding that the parabola has an eccentricity of 1 is critical as it simplifies the process of finding the polar equation from the given conditions.

Polar Equations

Polar equations offer a method to represent conic sections using polar coordinates, which include a radius \( r \) and an angle \( \theta \). This representation is particularly useful in scenarios where the focus of the conic section is located at the origin. The general polar equation of a conic section can be given as:\[ r = \frac{ed}{1 - e \sin \theta} \]where \( e \) is the eccentricity, \( d \) is the distance to the directrix, and \( \theta \) is the angle.For a parabola, with \( e = 1 \), the equation becomes\[ r = \frac{d}{1 - \sin \theta} \]In our example, where the directrix is at \( y = 2 \), we find \( d = 2 \) and thus the equation simplifies to:\[ r = \frac{2}{1 - \sin \theta} \]This equation effectively describes the set of all points \((r, \theta)\) that form the desired conic section, relative to its specified directrix and focus. Polar equations are not only crucial for understanding conic sections, but they also hold significant applications in physics and engineering, such as planetary motion and antenna design.