Problem 42 - College Algebra

Question

For the following exercises, graph the given conic section. If it is a parabola, label the vertex, focus, and directrix. If it is an ellipse, label the vertices and foci. If it is a hyperbola, label the vertices and foci. $$ r(6-4 \cos \theta)=5 $$

Step-by-Step Solution

Answer

The graph is an ellipse with vertices at (2.5, 0) and foci at (1.67, 0).

Step 1: Identify the conic section type

The given equation is in polar coordinates: \( r(6-4 \cos \theta) = 5 \). Rewrite it as \( r = \frac{5}{6 - 4 \cos \theta} \). This resembles the general conic form \( r = \frac{ed}{1 + e \cos \theta} \) or \( r = \frac{ed}{1 - e \cos \theta} \), indicating it is a conic section.

Step 2: Convert to standard conic form

The rewritten equation \( r = \frac{5}{6 - 4 \cos \theta} \) can be put into a more recognizable conic form of \( \frac{r}{1 - e \cos \theta} = \frac{ed}{1 - e \cos \theta} \). Here, \( e = \frac{4}{6} = \frac{2}{3} \), and it matches the form for an ellipse since \( e < 1 \).

Step 3: Calculate the lengths of the ellipse axes

The general form \( r = \frac{ed}{1 - e \cos \theta} \) indicates the ellipse's semi-major axis is \( d = \frac{5}{2} \) (the numerator of our original equation divided by \(\frac{6}{4}\)). The semi-minor axis length can be obtained via \( b = d\sqrt{1-e^2} \).

Step 4: Sketch the ellipse and label essential points

With \( a = \frac{5}{2} \) and \( b = \frac{5}{2} \times \sqrt{1-(\frac{2}{3})^2} \), plot these lengths along the axes regarding the orientation. Label the vertices \((d, 0)\), \((-d, 0)\), and the foci \((ed, 0)\) and \((-ed, 0)\) with \( e = \frac{2}{3} \).

Key Concepts

Polar Coordinates

Polar coordinates offer a way to express locations or points in a plane using two different values: the radial distance from a given reference point and the angle from a given reference direction. This is different from Cartesian coordinates, which use x and y axes.

When we talk about polar coordinates, we generally define them as \( (r, \theta) \). Here, \( r \) represents the distance from the origin or pole, while \( \theta \) denotes the angle measured from the positive x-axis. The origin is often called the pole, and the initial line (positive x-axis) is the polar axis.

• **Radial Distance (r)**: This is the length from the pole to a given point on the plane.
• **Angular Coordinate (\( \theta \))**: The angle that the line makes with the positive x-axis.

Using polar coordinates, you can represent many shapes, but it is especially useful for conic sections like ellipses, which have symmetric properties about a focal point.

Ellipse

An ellipse is a kind of conic section. It looks like a stretched circle. You can think of an ellipse as all the points where the sum of the distances to two fixed points is constant. These fixed points are known as the foci.

The equation for an ellipse can vary slightly between the Cartesian and polar coordinate systems. In polar coordinates, the ellipse is given by the equation \( r = \frac{ed}{1 - e \cos \theta} \), where \( e \) is the eccentricity (a measure of how much the ellipse deviates from being a circle).

• **Eccentricity (e)**: If \( e < 1 \), the conic is an ellipse. This means the ellipse is not a circle, but more elongated.
• **General Equation**: The standard form indicates an ellipse when the eccentricity \( e \) value lies between 0 and 1.

Ellipses are very common in physics and celestial mechanics, like the orbits of planets around a star.

Semi-Major Axis

The semi-major axis is one of the principal features of an ellipse. It is half of the longest diameter across the ellipse, which means it stretches from the center of the ellipse to the furthest point along the edge.

For the given ellipse equation in standard polar form, \( r = \frac{5}{6 - 4 \cos \theta} \), we determined a semi-major axis length of \( d = \frac{5}{2} \). The semi-major axis represents the radius of the largest circle that fits inside the ellipse.

• **Length**: It's crucial for calculating other properties of the ellipse, including area and perimeter.
• **Axis Extension**: This axis helps determine the longest expanse across the ellipse and shape orientation.

Understanding the semi-major axis is fundamental in grasping the overall geometry of ellipses and how they function in larger spaces.

Foci

The foci of an ellipse are the two significant points that define its shape. They are located along the major axis, symmetrically spaced from the center. The sum of the distances from any point on the ellipse to these two foci remains constant—a defining property of ellipses.

In the given solution, the ellipse in the polar form \( r = \frac{5}{6 - 4 \cos \theta} \) has eccentricity \( e = \frac{2}{3} \). This eccentricity helps locate the foci at positions \( (ed, 0) \) and \( (-ed, 0) \), which translates to the points \( (\frac{10}{6}, 0) \) and \( (-\frac{10}{6}, 0) \) in standard calculations.

• **Constant Distance**: The hallmark of an ellipse is how every point on it maintains a consistent total distance from the two foci.
• **Importance in Orbital Mechanics**: In celestial mechanics, the foci are where massive bodies like stars or planets are often located.

Knowing about the foci is vital for delving into deeper mathematical explanations and applications of ellipses in various scientific fields.