Problem 20 - Algebra and Trigonometry

Question

A polar equation of a conic is given. (a) Show that the conic is a parabola and sketch its graph. (b) Find the vertex and directrix and indicate them on the graph. $$ r=\frac{2}{5-5 \cos \theta} $$

Step-by-Step Solution

Answer

The conic is a parabola with vertex at \((\frac{2}{5}, 0)\) and directrix at \(r = 2\).

Step 1: Identify Eccentricity

In the polar equation for conics of the form \( r = \frac{ed}{1 - e \cos \theta} \), \( e \) is the eccentricity. In this case, rewrite the equation as \( r = \frac{2}{5(1 - 1 \cdot \cos \theta)} \). Here, \( e = 1 \), indicating a parabola.

Step 2: Verify the Equation Form

The polar equation \( r = \frac{2}{5 - 5 \cos \theta} \) matches the form \( r = \frac{ed}{1 - e \cos \theta} \) with \( e = 1 \), confirming that the conic is a parabola. A conic with \( e = 1 \) is always a parabola.

Step 3: Find the Vertex

For the parabola, the vertex is on the axis of symmetry, which in polar coordinates is at the pole (\( \theta = 0 \)). So the vertex is at \( (\frac{2}{5}, 0) \).

Step 4: Determine the Directrix

The directrix is located at \( r = \frac{d}{1-e} = 2 \) to the right since \( e = 1 \).

Step 5: Sketch the Graph

Draw the pole (origin), vertex at \( (\frac{2}{5}, 0) \), and the directrix as a line parallel to the y-axis at \( r = 2 \). The graph is a parabola opening towards the origin.

Key Concepts

Conic Sections

Conic sections are curves obtained by intersecting a plane with a double-napped cone. They consist of four distinct shapes: circles, ellipses, parabolas, and hyperbolas.
These sections are fundamental in geometry and occur in various applications such as astronomy, physics, and engineering.
Differences in shapes are determined by a particular value called eccentricity. Conic sections can be identified based on their eccentricity and geometric properties.

• A circle has an eccentricity of zero.
• An ellipse has an eccentricity between 0 and 1.
• A parabola has an eccentricity of exactly 1.
• A hyperbola has an eccentricity greater than 1.

Understanding these different sections can aid in the comprehension of the principles of conics.

Eccentricity

Eccentricity is a crucial concept that describes the shape of a conic section. It is denoted by the letter "e" and determines how much a conic section deviates from being circular.
In terms of conic sections in the polar form, eccentricity plays a vital role in defining their geometry.
Eccentricity values define each conic section as follows:

• Eccentricity of 0 results in a perfect circle.
• 0 < Eccentricity < 1 forms an ellipse, which is longest along its major axis.
• Eccentricity of 1 characterizes a parabola, where this exercise's focus lies.
• Eccentricity greater than 1 produces a hyperbola, split into two separate branches.

By recognizing the eccentricity value, one can anticipate the nature of the conic section, making it easier to sketch or analyze.

Parabola

A parabola is a unique conic section where all points are equidistant from a fixed point, called the focus, and a line, called the directrix.
The critical property that differentiates a parabola from other conic sections is its eccentricity, which is always 1.
Parabolas have a symmetrical, open curve shape, and they are often seen in real-life structures such as satellite dishes and bridges. Characteristics of a parabola include:

• The vertex, the point that represents the parabola's peak or lowest point, serves as its axis of symmetry.
• The directrix is a line perpendicular to the axis of symmetry and serves as a reference line for measurements.
• In polar coordinates, parabolas can be expressed with equations where the eccentricity, e, is 1, confirming their nature.

Understanding the fundamentals of a parabola can help in identifying its geometric properties when seen in equations or graphs.

Polar Coordinates

Polar coordinates provide an alternative framework to Cartesian coordinates, using a radial distance and angle for defining locations in a plane.
This system is highly valuable, especially when dealing with curves or regions that are naturally circular or symmetrical.
In polar coordinates, each point is represented by an ordered pair:

• The radial coordinate (r), which specifies the distance from the origin to the point.
• The angular coordinate (θ), which specifies the direction of the line from the origin to the point, measured in radians or degrees from the positive x-axis.

Conic sections like parabolas can be expressed simply and effectively in polar coordinates.
The polar equation for conics is often: \[ r = \frac{ed}{1 - e \cos \theta} \]where "e" is the eccentricity, and "d" is the distance constant. Polar coordinates provide unique insights and simplify complex geometric representations, making them vital in mathematical problem-solving and application.