Problem 62 - 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 \(A-H,\) identify the values of \(a^{2}\) and \(b^{2}\).

Step-by-Step Solution

Answer

Find the denominators in each equation for a² and b².

Step 1: Identify the Conic Section for Each Equation

First, determine whether each given equation is an ellipse, hyperbola, or circle by examining the relationship between the terms. For ellipses and circles, both terms are added, while for hyperbolas, the terms are subtracted.

Step 2: Determine Values of a² and b² for Ellipses and Circles

For equations in the form \( \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 \), you'll know the values of \( a^{2} \) and \( b^{2} \) are exactly the denominators. Circles will have \( a^{2} = b^{2} \).

Step 3: Determine Values of a² and b² for Hyperbolas

For equations in the form \( \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 \) or \( \frac{y^{2}}{b^{2}} - \frac{x^{2}}{a^{2}} = 1 \), identify \( a^{2} \) and \( b^{2} \) as the denominators in the equation, where \( a^{2} \) is associated with the positive term.

Key Concepts

Eccentricity

Eccentricity is a crucial concept when studying conic sections, as it helps to describe the shape and elongation of the paths these curves take. It is essentially a measure of how much a conic section deviates from being a circle. In mathematical terms, eccentricity is denoted by the letter "e." Here's how it applies to each conic section:

• For a circle, the eccentricity is 0, meaning it is a perfect circle with no elongation.
• For an ellipse, the eccentricity is greater than 0 but less than 1, indicating it is an elongated circle.
• For a hyperbola, the eccentricity is greater than 1, showing a significant deviation from a circular shape.

To calculate the eccentricity for an ellipse or a circle, use the formula:
\[ e = \frac{c}{d} \] where \( c^{2} = |a^{2} - b^{2}| \) and \( d \) is the larger of \( a \) or \( b \). For hyperbolas, the formula is similar, but \( c^{2} = a^{2} + b^{2} \) and \( d \) depends on the intercepts present. Understanding eccentricity helps us appreciate the diversity of orbits and paths described by these sections.

Hyperbola

A hyperbola is a fascinating conic section, represented as a curve formed by intersecting a double cone with a plane. It is defined by the equation: \[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \] or \[ \frac{y^2}{b^2} - \frac{x^2}{a^2} = 1 \] The equation's form indicates subtraction, which is a key identifying feature of hyperbolas. In a hyperbola, each branch of the curve opens either outward or downward, depending on the axis. This property makes it unique compared to ellipses and circles.
For hyperbolas, the eccentricity is greater than 1, showing a large deviation from a perfect circle. Calculating the eccentricity involves:

• Using the equation: \( e = \frac{c}{d} \) where \( c^2 = a^2 + b^2 \)
• Determining \( d \), which equals \( a \) if the hyperbola has \( x \)-intercepts and \( b \) if it has \( y \)-intercepts.

Hyperbolas have practical roles in phenomena like orbits and certain engineering applications.

Ellipse

An ellipse is another important type of conic section, resembling an elongated circle. An ellipse can be visualized by squeezing a circle along one axis. The formula for an ellipse is: \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \] The '+' sign in this equation differentiates it from a hyperbola. Ellipses are defined by two axes: the longer major axis and the shorter minor axis.
In astronomy, many orbits are elliptical, which relates to planetary motion and satellites. The eccentricity of an ellipse ranges between 0 and 1, which measures how much the shape deviates from being circular. Here’s how the eccentricity is calculated for an ellipse:

• The formula is \( e = \frac{c}{d} \) where \( c^2 = |a^2 - b^2| \)
• \( d \) is the larger value between \( a \) and \( b \)

Understanding ellipses is crucial for topics in both physics and geometry.

Circle

The circle is the simplest of the conic sections, characterized by its perfect roundness. A circle can be defined by the formula: \[ \frac{x^2}{r^2} + \frac{y^2}{r^2} = 1 \] where \( r \) represents the radius of the circle. The key feature of a circle is its eccentricity, which is always 0, indicating no elongation. No matter how you look at a circle, it remains uniformly distant from its center, which is a defining property of this shape.
Circles are commonly found in real-life applications, from wheels to full moons, due to their symmetry and simplicity. They play a significant role in both mathematics and nature. The symmetry of a circle makes its study straightforward, yet it still provides a foundation for understanding more complex conic sections like ellipses and hyperbolas.