Problem 70 - 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. \(\frac{x^{2}}{144}+\frac{y^{2}}{140}=1\) Graph each of the conic sections given in equations \(\mathrm{A}-\mathrm{H}\). What do you notice about the shape of the ellipses for increasing values of eccentricity? Which is the most elliptical? Which is the least elliptical, that is, the most circular?

Step-by-Step Solution

Answer

The ellipse with the highest eccentricity is most elliptical, while the lowest eccentricity ellipse is most circular. A circle has eccentricity 0.

Step 1: Identify Conic Section Type

First, determine the conic section type from each equation. Equations in the form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) represent ellipses if \(a\) and \(b\) are positive, while those in the form \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) represent hyperbolas.

Step 2: Calculate Eccentricity for Ellipses

For equations \(B\), \(C\), \(E\), \(F\), and \(H\) (ellipses), use the formula \(e = \frac{c}{d}\), where \(c^2 = |a^2 - b^2|\) and \(d\) is the larger of \(a\) or \(b\). Compute \(c\) and \(e\) for each equation.

Step 3: Calculate Eccentricity for Hyperbolas

For equations \(A\), \(D\), and \(G\) (hyperbolas), use the formula \(e = \frac{c}{d}\), where \(c^2 = a^2 + b^2\) and \(d\) is equal to \(a\) if the hyperbola has \(x\)-intercepts (Equation \(A\) and \(G\)), or \(b\) if the hyperbola has \(y\)-intercepts (Equation \(D\)). Calculate \(c\) and \(e\) for each.

Step 4: Determine Most Elliptical and Most Circular

Once the eccentricities are calculated, the ellipse with the highest eccentricity is the most elliptical, and the ellipse with the smallest eccentricity is the most circular. Compare the eccentricity values obtained for the ellipses.

Key Concepts

Eccentricity

In the world of conic sections, eccentricity is a critical concept used to describe the shape and uniqueness of various curves like circles, ellipses, and hyperbolas.
It is represented by the symbol \( e \) and is a measure of how much a conic section deviates from being perfectly circular.In simple terms, eccentricity helps us understand the extent to which these curves are stretched or compressed. For circles, the eccentricity is zero, indicating a perfect round shape.
For ellipses, the eccentricity ranges from zero to less than one, with values closer to zero indicating shapes more like circles. Hyperbolas have an eccentricity greater than one, indicating an open shape.Calculating eccentricity involves understanding the geometric properties of the conic section. In the case of ellipses and circles, eccentricity is computed using the formula:

• \( e = \frac{c}{d} \)

Here, \( c \) represents the distance from the center to the foci, while \( d \) is the length of the semi-major axis (the longest radius).
For hyperbolas, the calculations slightly differ due to their open curve nature. This formula provides insight into how the focal distance relates to the overall shape of the conic section.

Ellipse

An ellipse is a type of conic section that resembles a flattened circle.
It has two axes: the major axis and the minor axis. The major axis is the longest diameter, and the minor axis is the shortest.To visualize an ellipse, imagine stretching a circle along one direction.
The result is an elongated shape with two fixed points known as foci. The sum of the distances from any point on the ellipse to the foci remains constant.Ellipses are crucial in astronomy because they describe the orbits of celestial bodies like planets and stars. The ellipse equation is typically given by:

• \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \)

Where \( a \) and \( b \) are the radii along the x and y axes, respectively.
If \( a = b \), we have a circle, but usually, \( a > b \), indicating the elongated form.
As eccentricity increases from zero, the ellipse becomes increasingly elongated, whereas an eccentricity of zero designates a perfect circle.

Hyperbola

A hyperbola is another fascinating conic section with a unique, open shape formed by the intersection of a double cone with a plane.
Unlike ellipses, hyperbolas have two distinct branches that mirror each other.These branches open either horizontally or vertically, depending on the orientation.
The general equation for a hyperbola is:

• \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \)

In this form, \( a \) and \( b \) represent the distances from the center to the vertices and the co-vertices, respectively.
The asymptotes, which are lines that the hyperbola approaches but never touches, contribute to its unique appearance.Eccentricity of hyperbolas is always greater than one.
This high eccentricity value underlines their stretched nature, contrasting with the finite curves of ellipses and circles.
When astronomers discuss orbits with hyperbolic paths, they refer to objects like comets that pass through a system only once before heading back into space.