Problem 31 - Precalculus with Calculus Previews

Question

Identify the given rotated conic. Find the polar coordinates of its vertex or vertices. $$ r=\frac{4}{1+\cos (\theta-\pi / 4)} $$

Step-by-Step Solution

Answer

The conic is a parabola. The vertex in polar coordinates is at \((2, \pi/4)\).

Step 1: Identify the Conic Type

The given equation is of the form \( r = \frac{4}{1 + e \cos(\theta - \theta_0)} \). This form is a polar equation of a conic section, where \( e \) is the eccentricity of the conic. Here, \( e = 1 \), which means the conic is a parabola. In polar form, setting \( e=1 \) indicates the conic is a parabola.

Step 2: Determine the Direction of the Parabola

The equation \( r = \frac{4}{1 + \cos(\theta - \pi/4)} \) suggests the directrix is along the line \( \theta = \pi/4 \). Since the directrix is offset by \( -\pi/4 \) and \( \cos \) was used, this tells us the parabola opens towards \( \theta = -\pi/4 + 2k\pi \), effectively aligned at \( \theta = \pi/4 \).

Step 3: Identify the Vertex of the Parabola

Since this is a parabola in polar coordinates, the vertex can be found by considering where \( \theta \) matches the direction of the directrix variance. This vertex occurs at \( \theta = \theta_0 = \pi/4 \), and substituting in the main equation, the radial distance at the vertex \( r = \frac{4}{1 + \cos(0)} = \frac{4}{2} = 2 \). Therefore, the polar coordinates of the vertex are \( (2, \pi/4) \).

Key Concepts

Parabola

A parabola is one of the simplest forms of conic sections, and it's often described as the set of all points equidistant from a fixed point, called the **focus**, and a line, known as the **directrix**. In polar coordinates, parabolas have unique properties and representations.

• In the equation form, a parabola can typically be described by equations like \( r = \frac{l}{1 + \, e \, \cos(\theta - \theta_0)} \), where the eccentricity \( e = 1 \).
• The vertex of the parabola is the point on the parabola closest to the focus.
• They open in the direction defined by the angle \( \theta_0 \).

When dealing with polar coordinates, the equation reveals important characteristics of a parabola, including its orientation and vertex. For instance, the vertex point in polar coordinates is specifically the radial distance \( r \) when \( \theta = \theta_0 \). In our exercise, the parabola opens towards an angle \( \theta = \pi/4 \), with a vertex at \( (2, \pi/4) \). Understanding these traits helps us distinguish parabolas from other conic sections.

Eccentricity

Eccentricity is an integral attribute in classifying and understanding conic sections. It measures how much a conic section deviates from being circular. In polar coordinates, the eccentricity directly influences the shape and type of the conic.

• Circle: has an eccentricity \( e = 0 \). It’s perfectly round.
• Ellipse: (not a circle) has eccentricity \( 0 < e < 1 \).
• Parabola: has an eccentricity \( e = 1 \). It’s defined by having equal distance from the focus to the directrix.
• Hyperbola: has eccentricity \( e > 1 \).

For our exercise, the eccentricity \( e = 1 \) indicates the conic is a parabola. It aligns with the equation's form, verifying that in polar form, when \( e = 1 \), the parabola's characteristics are consistent across different points in the conic. This emphasizes the role eccentricity plays in shaping and understanding geometric properties of conics.

Conic Sections

Conic sections arise from the intersection of a plane and a cone- like shapes in geometry and physics. They encompass circles, ellipses, parabolas, and hyperbolas, each determined by the angle at which the plane intersects the cone.

• Circle: The plane cuts the cone parallel to its base.
• Ellipse: Similar to a circle, but the plane is tilted more such that it’s not parallel to the base.
• Parabola: The plane intersects the cone parallel to its slanting side, producing a distinct, U-shaped curve.
• Hyperbola: The plane cuts through both halves of the cone, forming two open curves.

In the context of polar coordinates, conic sections are described by equations that reveal their unique properties. For example, the polar form \( r = \frac{l}{1+e \cos(\theta-\theta_0)} \) indicates a conic's type and direction. This flexibility in representation allows for comprehensive analysis and applications in physics, engineering, and astronomy to model trajectories, orbits, and more. Understanding these concepts enriches one's ability to solve related mathematical problems.