Problem 45 - 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: \(y=2 ; e=2\)

Step-by-Step Solution

Answer

The polar equation of the conic is \( r = \frac{4}{1 + 2\sin\theta} \).

Step 1: Understand the Problem

We need to find the polar equation of a conic. The conic is defined by its eccentricity, \( e = 2 \), and its directrix, a line defined as \( y = 2 \). The focus of the conic is at the origin.

Step 2: Determine the Type of Conic

The eccentricity \( e = 2 \) indicates that the conic is a hyperbola, since hyperbolas have eccentricities \( e > 1 \).

Step 3: Set Up the Polar Equation Form

The general polar equation of a conic is given by \( r = \frac{ed}{1 \, + \, e \sin \theta} \), where \( d \) is the perpendicular distance from the origin to the directrix. Since the directrix is \( y = 2 \), thus \( d = 2 \).

Step 4: Substitute Values into the Equation

Substitute \( e = 2 \) and \( d = 2 \) into the equation, obtaining \[ r = \frac{4}{1 + 2\sin\theta} \]. This is the polar equation of the conic.

Key Concepts

Eccentricity

Eccentricity is a fundamental concept in the study of conic sections. It helps to describe the shape and nature of a conic section and is denoted by the letter \( e \). Depending on the value of \( e \), we can determine the type of conic section:

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

Understanding eccentricity is crucial because it allows us to classify conic sections and predict their properties. For example, in this exercise, an eccentricity of \( e = 2 \) signifies the conic is a hyperbola. Eccentricity also relates to how "stretched out" or "elongated" a conic section is in comparison to a circle. The greater the eccentricity, the more elongated the conic section is.

Conic Sections

Conic sections are curves obtained by intersecting a plane with a double-napped cone. The four primary types of conic sections are circles, ellipses, parabolas, and hyperbolas. These shapes are essential in various fields including geometry, astronomy, and physics. The type of conic section formed depends on the angle at which the plane intersects the cone.

• A circle is formed when the intersection is perpendicular to the axis of the cone.
• An ellipse occurs when the plane cuts through the cone at an angle but does not intersect the base.
• A parabola is created when the plane is parallel to a generating line of the cone.
• A hyperbola is formed when the plane intersects both nappes of the cone.

The study of conics includes understanding their algebraic equations, properties, and graphs. In polar coordinates, conics with their focus at the origin are particularly significant as they provide a unique way to describe these curves using equations involving eccentricity and directrix.

Hyperbola

A hyperbola is a conic section that appears as two separate curves known as branches. These branches mirror each other and are characterized by their openness. The term "hyperbola" is derived from the Greek word meaning "over-thrown" or "excess," highlighting its property of having an eccentricity greater than one.The defining equation of a hyperbola in polar coordinates with the focus at the origin and the directrix parallel to the axis is given by:\[ r = \frac{ed}{1 + e\sin\theta} \text{ or } r = \frac{ed}{1 + e\cos\theta} \]Here, \( e \) is the eccentricity and \( d \) is the perpendicular distance from the origin to the directrix. Hyperbolas have several unique properties:

• They have two foci and are symmetrical with respect to these foci.
• Each branch asymptotically approaches two lines, called the asymptotes.
• The transverse axis connects the vertices of the hyperbola, while the conjugate axis is perpendicular to it.

In practical applications, hyperbolas can be seen in navigation systems and the paths of certain celestial bodies. Their mathematical properties make hyperbolas an integral shape not just in geometry, but in real-world applications as well.