Problem 67 - Intermediate Algebra

Question

The orbits of stars, planets, comets, asteroids, and satellites all have the shape of one of the conic sections. Astronomers use a measure called eccentricity to describe the shape and elongation of an orbital path. For the circle and ellipse, eccentricity e is calculated with the formula \(e=\frac{c}{d},\) where \(c^{2}=\left|a^{2}-b^{2}\right|\) and \(d\) is the larger value of a or b. For a hyperbola, eccentricity e is calculated with the formula \(e=\frac{c}{d},\) where \(c^{2}=a^{2}+b^{2}\) and the value of \(d\) is equal to a if the hyperbola has \(x\) -intercepts or equal to b if the hyperbola has \(y\) -intercepts. A. \(\frac{x^{2}}{36}-\frac{y^{2}}{13}=1\) B. \(\frac{x^{2}}{4}+\frac{y^{2}}{4}=1\) C. \(\frac{x^{2}}{25}+\frac{y^{2}}{16}=1\) D. \(\frac{y^{2}}{25}-\frac{x^{2}}{39}=1\) G. \(\frac{x^{2}}{16}-\frac{y^{2}}{65}=1\) E. \(\frac{x^{2}}{17}+\frac{y^{2}}{81}=1\) F. \(\frac{x^{2}}{36}+\frac{y^{2}}{36}=1\) H. \(\frac{x^{2}}{144}+\frac{y^{2}}{140}=1\) What do you notice about the values of \(e\) for the equations you identified as circles?

Step-by-Step Solution

Answer

The eccentricity values for circles are 0.

Step 1: Identify the Type of Conic Section for Each Equation

For each equation, determine whether it is an ellipse or a hyperbola. An equation of the form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) or \(\frac{y^2}{a^2} + \frac{x^2}{b^2} = 1\) represents an ellipse. If the sign between the terms is negative, as in \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), then it represents a hyperbola.

Step 2: Calculate Eccentricity for Ellipses and Circles

For the equations identified as ellipses \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\): 1. Compute \(c^2 = |a^2 - b^2|\).2. Determine \(d\) as the larger of \(a\) or \(b\).3. Calculate eccentricity \(e = \frac{c}{d}\). For circles, where \(a = b\), \(e\) will be zero because \(c=0\).

Step 3: Calculate Eccentricity for Hyperbolas

For equations identified as hyperbolas:1. Compute \(c^2 = a^2 + b^2\).2. Determine \(d = a\) if the equation is of the form \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), or \(d = b\) if the equation is of the form \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\).3. Calculate eccentricity \(e = \frac{c}{d}\).

Step 4: Identify Circles and Calculate Their Eccentricity

A circle is a special type of ellipse where \(a = b\), which means there is no elongation, and the eccentricity \(e = 0\). Examine the equations:- Equations F and B, \(\frac{x^2}{36} + \frac{y^2}{36} = 1\) and \(\frac{x^2}{4} + \frac{y^2}{4} = 1\), represent circles because their denominators are equal.

Step 5: Conclude About Eccentricity for Circles

For equations F and B:- Since they represent circles and have the form \(\frac{x^2}{r^2} + \frac{y^2}{r^2} = 1\), the eccentricity \(e = \frac{c}{d} = 0\) because \(c = 0\) due to \(a = b\).

Key Concepts

Eccentricity

Eccentricity is a key concept when discussing conic sections. It measures how much a conic section deviates from being circular. In simple terms, eccentricity tells us whether the shape is stretched out or more rounded.

Here's how eccentricity works for different conic sections:

• For circles, the eccentricity is zero because circles are perfectly round.
• For ellipses, the eccentricity is between 0 and 1, indicating a stretched oval shape.
• For hyperbolas, the eccentricity is greater than 1, which shows that they have an open and elongated shape.

When you calculate eccentricity, you use the formula:\[e = \frac{c}{d},\]where \(c\) is a measure of the distance between the foci (focal points), and \(d\) is the semi-major axis or a value related to the hyperbola's intercepts, depending on the context.

Understanding eccentricity can help us gauge the shape of orbits of celestial bodies like planets and comets, as astronomical orbits are often described in terms of this parameter.

Ellipse

An ellipse is one of the types of conic sections. Think of it as a stretched or elongated circle. The sun rises as a point in the center if we're depicting planetary orbits as ellipses. In an ellipse:

• The longest diameter is called the semi-major axis.
• The shortest diameter is the semi-minor axis.
• The sum of distances from any point on the ellipse to the two focal points is constant.

Mathematically, an ellipse is represented as:\[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\]where \(a\) and \(b\) are the lengths of the semi-major and semi-minor axes respectively.

In nature, many planetary orbits are shaped as ellipses, thanks to gravitational forces. Understanding how eccentricity applies to ellipses allows scientists to predict seasons and orbital paths effectively.

Hyperbola

Hyperbolas are another fascinating type of conic section. Unlike circles and ellipses, a hyperbola consists of two mirrored curves that extend infinitely. These curves are often used to describe certain natural and manmade phenomena. In mathematics, a standard hyperbola has the formula:

\[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\]for those opening along the x-axis. Or:\[\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\]for those opening along the y-axis.

Critical features of hyperbolas include:

• Two branches that never meet.
• Two foci, located outside the curves.
• The difference in distances from any point on the curve to the foci is consistent.

Scientists often observe hyperbolic paths in the flight paths of spacecraft or the curves traced by light rays in certain lenses. The eccentricity of a hyperbola is greater than 1, meaning it stretches out more than an ellipse.

Circle

A circle is perhaps the most straightforward conic section to visualize. It is defined as a set of points in a plane that are equidistant from a center point. Because of its symmetry, all diameters of a circle are equal, reinforcing the idea of a perfect round shape. The equation for a circle is:\[\frac{x^2}{r^2} + \frac{y^2}{r^2} = 1\]where \(r\) is the radius.

• All points on a circle are the same distance from the center.
• Eccentricity \(e\) for a circle is always 0.
• The circle is a special case of an ellipse where the semi-major and semi-minor axes are equal.

Understanding circles' properties aids in grasping more complex shapes like ellipses and hyperbolas. Even though celestial orbits are rarely perfect circles, this concept helps set the groundwork for studying other conic sections.