Problem 22 - Precalculus with Calculus Previews

Question

Find a polar equation of the conic with focus at the origin that satisfies the given conditions. $$ e=1, \text { directrix } y=-2 $$

Step-by-Step Solution

Answer

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

Step 1: Understand the Problem

We need to find a polar equation of a conic with the eccentricity \( e = 1 \) and the directrix \( y = -2 \). The focus of the conic is at the origin.

Step 2: Identify the Conic Type

Since the eccentricity \( e = 1 \), the conic is a parabola. Parabolas have the property that the focus and directrix are equidistant from any point on the conic.

Step 3: Formula For Parabolas in Polar Coordinates

For a conic section with the focus at the origin and a horizontal directrix \( y = k \), the general polar equation is \( r = \frac{ke}{1 + e \sin \theta} \) when the directrix is below the focus.

Step 4: Substitute Known Values

Here, \( e = 1 \) and the directrix equation is \( y = -2 \) which implies \( k = -2 \). Substitute into the formula: \[ r = \frac{-2 \cdot 1}{1 + 1 \sin \theta} = \frac{-2}{1 + \sin \theta} \].

Step 5: Final Simplified Polar Equation

Since \( e = 1 \), the equation becomes \[ r = \frac{-2}{1 + \sin \theta} \]. This is the polar equation of the conic section which is a parabola with focus at the origin and directrix at \( y = -2 \).

Key Concepts

Conic Sections

Conic sections are curves that can be formed by the intersection of a plane and a double-napped cone. These come in several types, but the most commonly known types are:

• Circles - Formed when the intersection is perpendicular to the cone's axis.
• Ellipses - Produced when the intersection is at an angle, but not steep enough to form a hyperbola.
• Parabolas - Occur when the intersecting plane is parallel to the slope of the cone.
• Hyperbolas - Arise when the intersection is steep enough to cross both naps of the cone.

Each type of conic section has its own unique equation and properties. These sections are not only intriguing mathematically, but they also appear in various practical applications, ranging from architecture to astronomy. In polar coordinates, these sections can be defined using formulas that incorporate the eccentricity, focus, and directrix. This allows mathematicians and scientists to model and analyze curves in ways that are often more convenient than other coordinate systems. Understanding conic sections is crucial for studying diverse phenomena, from planetary orbits to the design of certain optical devices.

Eccentricity

Eccentricity is a measure used to describe the "shape" of a conic section. It tells us how much a conic section deviates from being a perfect circle.

• If the eccentricity, denoted as \( e \), is zero, the conic is a circle.
• When \( e \) is between 0 and 1, the shape is an ellipse.
• If \( e = 1 \), the conic is a parabola.
• For \( e > 1 \), the conic is a hyperbola.

The value of eccentricity not only decides the type of conic section but also its properties. For example, if a satellite's path around Earth has an eccentricity of zero, it would form a perfect circle, which would be highly unusual. For the exercise we discussed, the eccentricity \( e = 1 \) indicates that the shape of the conic section is a parabola. This means each point on the parabola is equidistant from a specific point, the focus, and a line, called the directrix. Eccentricity is a critical concept in understanding both their mathematics and the natural phenomena involving them.

Parabolas

Parabolas are unique conic sections with a distinct shape and properties. They are defined as the set of all points that are equidistant from a point, known as the focus, and a line, called the directrix.

• In this context, the polar equation of a parabola helps simplify and represent its structure.
• For parabolas, the eccentricity is always equal to 1.

In polar coordinates, parabolas possess a specific equation format particularly when the directrix is horizontal. The equation used is: \[ r = \frac{ke}{1 + e \sin \theta} \]where \( e \) is the eccentricity, and \( k \) correlates to the directrix's position. In the original problem, the polar equation that resulted represents all points which keep the same distance from the focus at the origin and the line \( y = -2 \).
Parabolas are found in many real-world applications like satellite dishes and antennae designs because of their reflective properties. These properties highlight the importance of understanding parabolas beyond a purely mathematical perspective.