Problem 65 - 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. \(\frac{x^{2}}{144}+\frac{y^{2}}{140}=1\) For each of the equations \(\mathrm{A}-\mathrm{H}\), calculate the eccentricity \(e .\)

Step-by-Step Solution

Answer

Eccentricities: A) 7/6, B) 0, C) 3/5, D) 8/5, E) 8/9, F) 0, G) 9/4, H) 1/6.

Step 1: Identify Conic Section Type - Equation A

The equation for A is \(\frac{x^{2}}{36} - \frac{y^{2}}{13} = 1\). This is a hyperbola because it is in the form \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\).

Step 2: Calculate Eccentricity - Equation A

For hyperbolas, \(c^2 = a^2 + b^2\). Here, \(a^2=36\) and \(b^2=13\), so \(c^2 = 36 + 13 = 49\), giving \(c=7\). The hyperbola has \(x\)-intercepts, so \(d=a=6\). Eccentricity \(e = \frac{c}{d} = \frac{7}{6}\).

Step 3: Identify Conic Section Type - Equation B

The equation for B is \(\frac{x^{2}}{4} + \frac{y^{2}}{4} = 1\). This is a circle because \(a=b\).

Step 4: Calculate Eccentricity - Equation B

For a circle, \(e = 0\) because \(c = 0\) (as \(a = b\)).

Step 5: Identify Conic Section Type - Equation C

The equation for C is \(\frac{x^{2}}{25} + \frac{y^{2}}{16} = 1\). This is an ellipse because it is in the form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\).

Step 6: Calculate Eccentricity - Equation C

For ellipses, \(c^2 = |a^2 - b^2|\). Here, \(a^2=25\), \(b^2=16\). \(c^2 = 9\), so \(c=3\). Here, \(d=5\). Eccentricity \(e = \frac{3}{5}\).

Step 7: Identify Conic Section Type - Equation D

The equation for D is \(\frac{y^{2}}{25} - \frac{x^{2}}{39} = 1\). This is a hyperbola because it is in the form \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\).

Step 8: Calculate Eccentricity - Equation D

For hyperbolas, \(c^2 = a^2 + b^2\). Here, \(a^2=25\) and \(b^2=39\), so \(c^2 = 64\), giving \(c=8\). The hyperbola has \(y\)-intercepts, so \(d=a=5\). Eccentricity \(e = \frac{8}{5}\).

Step 9: Identify Conic Section Type - Equation E

The equation for E is \(\frac{x^{2}}{17} + \frac{y^{2}}{81} = 1\). This is an ellipse because it is in the form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\).

Step 10: Calculate Eccentricity - Equation E

For ellipses, \(c^2 = |b^2 - a^2|\). Here, \(a^2=17\), \(b^2=81\). \(c^2 = 64\), so \(c=8\). Here, \(d=9\). Eccentricity \(e = \frac{8}{9}\).

Step 11: Identify Conic Section Type - Equation F

The equation for F is \(\frac{x^{2}}{36} + \frac{y^{2}}{36} = 1\). This is a circle because \(a = b\).

Step 12: Calculate Eccentricity - Equation F

For a circle, \(e = 0\) because \(c = 0\) (as \(a = b\)).

Step 13: Identify Conic Section Type - Equation G

The equation for G is \(\frac{x^{2}}{16} - \frac{y^{2}}{65} = 1\). This is a hyperbola because it is in the form \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\).

Step 14: Calculate Eccentricity - Equation G

For hyperbolas, \(c^2 = a^2 + b^2\). Here, \(a^2=16\) and \(b^2=65\), so \(c^2 = 81\), giving \(c=9\). The hyperbola has \(x\)-intercepts, so \(d=a=4\). Eccentricity \(e = \frac{9}{4}\).

Step 15: Identify Conic Section Type - Equation H

The equation for H is \(\frac{x^{2}}{144} + \frac{y^{2}}{140} = 1\). This is an ellipse because it is in the form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\).

Step 16: Calculate Eccentricity - Equation H

For ellipses, \(c^2 = |a^2 - b^2|\). Here, \(a^2=144\), \(b^2=140\). \(c^2 = 4\), so \(c=2\). Here, \(d=12\). Eccentricity \(e = \frac{2}{12} = \frac{1}{6}\).

Key Concepts

Conic Sections

Conic sections are the curves obtained by intersecting a cone with a plane. There are four primary types of conic sections: circles, ellipses, parabolas, and hyperbolas. Each of these conic sections has a unique set of characteristics and can be represented by different algebraic equations. In the context of celestial orbits, conic sections are essential since the path of an orbiting body like a planet or asteroid is often a conic section.

Understanding the properties of conic sections helps astronomers describe and predict the motion of these celestial bodies. The shape of the path is influenced by providing eccentricity, a parameter that measures the deviation of a conic section from being a perfect circle.

By studying the conic section equations, we can quickly determine the type of conic and calculate its eccentricity, which directly influences the nature of the orbit.

Hyperbola

A hyperbola is a type of conic section that is formed when a plane cuts through both halves of a double cone. The resulting curve is two separate branches, which are mirror images of each other. In standard form, the equation of a hyperbola looks like \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\] for a hyperbola open along the x-axis, or \[\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\] for a hyperbola open along the y-axis.

The eccentricity of a hyperbola is always greater than 1. It's calculated using the formula: \[e = \frac{c}{d}\] where \(c^2 = a^2 + b^2\) and \(d\) corresponds to \(a\) or \(b\) depending on the axis the hyperbola is opened on.

Hyperbolas have important applications in physics and engineering, including in the paths of objects under certain influences affecting their trajectories.

Ellipse

An ellipse is a conic section that looks like a stretched-out circle. It results when a plane cuts through a cone at an angle that is neither parallel to the base nor the side. The general equation for an ellipse can be represented as:\[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\] if \(a^2 \geq b^2\), otherwise the axes are switched.

The eccentricity of an ellipse, denoted as \(e\), measures how much the ellipse deviates from being a perfect circle. It is calculated using the formula: \[e = \frac{c}{d}\] where \(c^2 = |a^2 - b^2|\) and \(d\) is the larger of \(a\) or \(b\). The eccentricity of an ellipse is always between 0 and 1.

Ellipses are significant in astronomy because many celestial bodies, such as planets, have elliptical orbits due to the gravitational forces involved.

Circle

A circle is the simplest form of a conic section and is defined as the set of all points in a plane that are equidistant from a fixed point called the center. The standard equation for a circle, where the center is at the origin, is:\[\frac{x^2}{a^2} + \frac{y^2}{a^2} = 1\] implying that \(a = b\).

The eccentricity of a circle is 0, because in the formula \(e = \frac{c}{d}\), \(c\) is 0 since \(a = b\). This indicates that a circle is a perfect round shape without elongation or deviation.

In geometry, circles are considered a special type of ellipse where the semi-major and semi-minor axes are equal, resulting in no flattening at all. This property of zero eccentricity makes circles unique among the conic sections.