Problem 5 - Precalculus: Mathematics 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: Understanding the Relationship

A parabola in polar coordinates with focus at the origin has the form \( r = \frac{ed}{1-e\sin\theta} \), where \( e \) is the eccentricity and \( d \) is the distance from the focus to the directrix. For a parabola, the eccentricity, \( e \), is 1.

Step 2: Identifying Directrix Distance

Given the directrix is \( y=2 \), the straight-line distance from the focus at the origin to the nearest point on this directrix (a horizontal line) is 2. Therefore, our \( d \) is 2.

Step 3: Formulate the Polar Equation

Substitute \( e = 1 \) and \( d = 2 \) into the standard form of the polar equation: \( r = \frac{2}{1 - 1\sin\theta} \). Simplify it to get the equation \( r = \frac{2}{1 - \sin\theta} \).

Key Concepts

Conic Sections

Conic sections are a fascinating topic in mathematics, linked to the study of slices from a cone. Imagine cutting a cone at different angles and you will discover the different sections: circles, ellipses, parabolas, and hyperbolas. Together, these shapes are referred to as conic sections. What makes conic sections unique is their geometric definitions and relationships between various coordinate systems, such as Cartesian and polar coordinates.

Understanding conic sections is critical for solving many geometrical problems. Each conic section has a distinct set of properties:

• Circle: A set of points equidistant from a central point.
• Ellipse: A set of points where the sum of distances from two foci is constant.
• Parabola: A set of points equidistant from a point (focus) and a line (directrix).
• Hyperbola: A set of points where the difference of distances from two foci is constant.

Conic sections are not only theoretical but also practical, as they appear in various real-world applications such as planetary orbits, lenses, and even architectural structures.

Parabola

A parabola is a special type of conic section that you can visualize as the set of all points that are equidistant from a point called the focus, and a straight line known as the directrix. In the real world, you'll find parabolas in places like satellite dishes and the arches of bridges.

In polar coordinates, a parabola has a unique representation. When its focus is at the origin, its equation is usually given as:\[ r = \frac{ed}{1 - e\sin\theta}\]By definition, the eccentricity \( e \) of a parabola is always 1. In the context of the exercise provided, the directrix is at \( y=2 \), with a calculated distance \( d \) of 2 from the focus to the directrix. Thus, substituting these into the polar equation gives:\[ r = \frac{2}{1 - \sin\theta}\]Understanding the behavior of a parabola helps in various technological applications, including the design of reflective surfaces that can focus light and sound.

Eccentricity

Eccentricity is a number that describes the shape of a conic section. It's a key to understanding how a conic looks and behaves. The eccentricity \( e \) plays a major role in defining the type of conic section:

• Circle: \( e = 0 \)
• Ellipse: \( 0 < e < 1 \)
• Parabola: \( e = 1 \)
• Hyperbola: \( e > 1 \)

For a parabola, the eccentricity is exactly 1. This characteristic defines its symmetrical open shape, and the fact that every point is equidistant from the directrix and focus. Certain equations or expressions in polar coordinates can be used to emphasize these properties. As seen in the solution, it allows the formation of a precise relationship in the polar equation:\[ r = \frac{2}{1 - \sin\theta}\]Eccentricity not only helps us distinguish conics but also aids in many engineering and physics problems, especially in understanding orbits and trajectories.