Problem 63 - 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 \(A-H,\) calculate the value of \(c^{2}\) and \(c .\)

Step-by-Step Solution

Answer

Calculate \(c^2\) using the formulas for ellipses and hyperbolas, then find \(c\) as the square root of \(c^2\) for each equation.

Step 1: Identify Conic Section Types

Determine whether each given equation represents an ellipse or a hyperbola. An equation is an ellipse if both terms are added and a hyperbola if one term is subtracted.

Step 2: Calculate \(c^2\) for Ellipses

For ellipses, where the equation format is \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), calculate \(c^2 = |a^2 - b^2|\). Substitute the values of \(a^2\) and \(b^2\) from the equation to find \(c^2\).

Step 3: Calculate \(c^2\) for Hyperbolas

For hyperbolas, where the equation format is either \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) or \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\), calculate \(c^2 = a^2 + b^2\). Use the respective denominators of the \(x^2\) and \(y^2\) terms as \(a^2\) and \(b^2\).

Step 4: Determine \(c\) Values

Calculate \(c\) by taking the square root of \(c^2\) for each conic section. Use the formula \(c = \sqrt{c^2}\) and ensure to simplify where possible.

Step 5: Apply Steps to Each Equation

Go through each equation from A to H, applying the steps above one by one to determine the corresponding \(c^2\) and \(c\).

Key Concepts

Eccentricity

In the study of conic sections, eccentricity plays a key role. This mathematical term is used to describe how much a conic section deviates from being a perfect circle. It is denoted by the variable \( e \).

• For a circle, the eccentricity is \( 0 \), as every point is equidistant from the center.
• For ellipses, \( 0 < e < 1 \). The closer \( e \) is to 0, the more circular the ellipse, and the closer \( e \) is to 1, the more elongated it appears.
• Hyperbolas have an eccentricity greater than 1, indicating their distinct open curves.
• For parabolas, the eccentricity is exactly 1, due to their characteristic symmetric curve.

For ellipses and hyperbolas, eccentricity is calculated as \( e = \frac{c}{d} \). This involves finding \( c \), which depends on either the sum or the difference of the squares of the semi-major and semi-minor axes, and \( d \), which is the length of the semi-major axis. A thorough understanding of eccentricity can give us insight into the shape and behavior of celestial orbits.

Ellipses

An ellipse is one type of conic section formed by a plane intersecting a cone. It's a closed curve and looks like a squished circle.The standard form of the equation for an ellipse is \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), where \( a \) and \( b \) are the semi-major and semi-minor axes, respectively. To calculate \( c^2 \) in an ellipse, use the formula:\[c^2 = |a^2 - b^2|\]- If \( a > b \), the ellipse is elongated horizontally. - If \( b > a \), it is elongated vertically. The concept of ellipses is especially useful in analyzing celestial paths. For instance, the orbits of planets around the sun are elliptical, not circular. The eccentricity of these orbits tells us how stretched they are. Kepler's First Law states, "The orbit of every planet is an ellipse with the sun at one of the two foci." This law highlights the elegance of ellipses in astronomy.

Hyperbolas

Hyperbolas are another kind of conic section characterized by their open, divergent curves. Similar to ellipses, they are formed through the intersection of a plane with a double cone; however, the angle and position of the plane result in a distinctly different shape.The standard equation for a hyperbola is \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) or \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \). Here, the vertices of the hyperbola help form the axes of symmetry, differing from ellipses which are always closed.To find \( c^2 \) for a hyperbola, use:\[c^2 = a^2 + b^2\]- If the hyperbola opens horizontally, \( a \) relates to \( x \).- If it opens vertically, \( a \) relates to \( y \).Hyperbolas occur frequently in physics and astronomy, such as in the orbit of some comets which potentially escape the Sun’s gravitational pull. Due to their unique shape, hyperbolas are employed in navigation systems and understanding asymptotic behavior in graphs.

Orbit Calculations

Orbits in space can be complex, but with knowledge of conic sections, these become more understandable. In astronomy, calculations involving orbits lean heavily on concepts from conic sections, particularly using ellipses and hyperbolas.Orbit calculations involve:

• Determining the type of orbit: elliptical, hyperbolic, or circular.
• Finding the eccentricity, which informs us about how circular or elongated the orbit is.
• Calculating \( c^2 \) and \( c \) to determine the shape and position of the focal points.

For planets and most satellites, elliptical orbits are typical due to gravitational forces. Knowing eccentricity helps calculate the perihelion (closest point) and aphelion (farthest point) distances in an orbit. Each orbital path tells a story about the celestial body's journey around its focal point, giving astronomers insights into its motion and gravitational influences.These calculations are essential for understanding not just the movement of planets, but also for missions involving spacecraft and satellites. They let us predict trajectories, ensuring successful transits and alignments with their intended destinations.