Problem 66 - 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 ellipses?

Step-by-Step Solution

Answer

Ellipses have eccentricity values less than 1.

Step 1: Identify Type of Conic Section

For each equation, determine if it represents a circle, ellipse, or hyperbola based on the signs in the equation. If both terms are positive, it is a circle or an ellipse; if one term is negative, it is a hyperbola. The specific form helps in further classification.

Step 2: Determine Circle, Ellipse or Hyperbola

A: Hyperbola, since one term is negative. B: Circle, since both terms are positive and coefficients are equal. C: Ellipse, since both terms are positive but coefficients differ. D: Hyperbola, since one term is negative. E: Ellipse, since both terms are positive but coefficients differ. F: Circle, since both terms are positive and coefficients are equal. G: Hyperbola, since one term is negative. H: Ellipse, since both terms are positive but coefficients differ.

Step 3: Calculate Eccentricity for Hyperbolas

For hyperbolas A and G:- A: - Here, \(a^2 = 36\), \(b^2 = 13\), so \(c^2 = 36 + 13 = 49\), hence \(c = 7\). Since it has \(x\)-intercepts, \(d = a = 6\). - Eccentricity, \(e = \frac{7}{6}\).- G: - Here, \(a^2 = 16\), \(b^2 = 65\), so \(c^2 = 16 + 65 = 81\), hence \(c = 9\). It also has \(x\)-intercepts, \(d = a = 4\). - Eccentricity, \(e = \frac{9}{4}\).

Step 4: Calculate Eccentricity for Ellipses

For ellipses C, E, and H:- C: - Here, \(a^2 = 25\), \(b^2 = 16\), so \(c^2 = 25 - 16 = 9\), hence \(c = 3\). The larger is \(a = 5\). - Eccentricity, \(e = \frac{3}{5}\).- E: - Here, \(a^2 = 17\), \(b^2 = 81\), so \(c^2 = 81 - 17 = 64\), hence \(c = 8\). The larger is \(b = 9\). - Eccentricity, \(e = \frac{8}{9}\).- H: - Here, \(a^2 = 144\), \(b^2 = 140\), so \(c^2 = 144 - 140 = 4\), hence \(c = 2\). The larger is \(a = 12\). - Eccentricity, \(e = \frac{1}{6}\).

Step 5: Conclude Observations on Ellipse Eccentricity

For ellipses C, E, and H, the eccentricities are calculations that range between 0 and 1. Specifically, they are \( \frac{3}{5}, \frac{8}{9}, \frac{1}{6} \) respectively.

Key Concepts

Conic Sections

Conic sections are fundamental curves that arise from the intersection of a plane and a cone. These sections describe the paths of celestial bodies and include diverse shapes such as circles, ellipses, hyperbolas, and parabolas. While this exercise focuses on circles, ellipses, and hyperbolas, it's intriguing how each of these shapes manifests in real-world celestial orbits. Conic sections are primarily classified based on the angle at which the plane intersects the cone. This results in different curves with unique properties.

• **Circle**: Obtained when the plane is perpendicular to the cone's axis.
• **Ellipse**: Derived when the plane cuts through the cone at an angle, not steep enough to create a parabola.
• **Hyperbola**: Formed by slicing the cone with a plane that intersects both nappes, the two rounded parts of the cone.

Each shape has its formula for calculating eccentricity, a key measure in understanding their geometrical properties. This measure is pivotal in identifying how the shape deviates from being circular. Understanding these concepts allows astronomers to detail the pathways of stars, planets, comets, and more.

Ellipse

An ellipse is an elongated circle. It emerges when a plane intersects a cone at an angle that is lesser than that for a circle.
Ellipses have two foci, and their defining characteristic is that the sum of the distances from any point on the ellipse to the two foci is constant. This property makes ellipses favorable in describing planetary orbits, as planets tend to travel in paths close to this shape.
The eccentricity of an ellipse (\( e = \frac{c}{d} \)) helps in determining how much the ellipse deviates from a perfect circle. In our exercise with equations such as \( \frac{x^2}{25} + \frac{y^2}{16} = 1 \) (Ellipse C), the eccentricity values are less than 1, indicating a more "stretched" circle. The value of eccentricity provides insights into how elongated the path is which directly influences orbital speeds relative to different planetary positions in its orbit.

Hyperbola

A hyperbola results from a plane cutting through both nappes of a double cone. This curve resembles an open pair of opposing curves. Hyperbolas appear in the context of space when describing certain celestial trajectories, such as those of objects that escape into space rather than maintaining a closed orbit.
In terms of eccentricity, hyperbolas exhibit an eccentricity greater than 1, indicating a diverging path rather than a closed loop. For instance, in equation \( \frac{x^2}{16} - \frac{y^2}{65} = 1 \) (Hyperbola G), the eccentricity is calculated to be greater than 1 (\( e = \frac{9}{4} \)). This value suggests a path that is more stretched compared to ellipses and circles.
Hyperbolas are applied in technology, such as the paths of radio waves and satellite dish reflectors, further emphasizing their significant presence beyond just astronomical applications.

Circle

A circle is a special case of an ellipse where both foci coincide at the same point. This makes the circle a perfect round shape, achieved by intersecting the cone perpendicular to its axis. In the given exercise, circles are represented by equations like \( \frac{x^2}{4} + \frac{y^2}{4} = 1 \) (Equation B), where the coefficients are equal, indicating uniformity in all directions.
The eccentricity of a circle is uniquely 0, showing there is no deviation from the center in any direction. This uniformity is what distinguishes circles from ellipses in conic sections.
In nature, perfect circular orbits are rare because of gravitational influences, but they are often used as approximations in initial orbit studies or in defining boundaries and limits of circular motion in both physical and theoretical frameworks. Circles provide a basis for understanding more complex conic sections like ellipses and hyperbolas.