Problem 61 - 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. \(\frac{x^{2}}{144}+\frac{y^{2}}{140}=1\) Identify the type of conic section represented by each of the equations \(\mathrm{A}-\mathrm{H}\)

Step-by-Step Solution

Answer

A: hyperbola; B: circle; C: ellipse; D: hyperbola; G: hyperbola; E: ellipse; F: circle; H: ellipse.

Step 1: Recognize Format of Each Equation

Identify if the equation is in the form of a conic section. For equations like \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) or \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \), it describes a hyperbola. For equations of the form \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), it describes an ellipse or a circle, depending on the equality of \(a\) and \(b\).

Step 2: Determine If it's a Circle or Ellipse

For equations of the form \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), if \( a^2 = b^2 \) then it's a circle. If not, it’s an ellipse. For example, \( \frac{x^2}{4} + \frac{y^2}{4} = 1 \) is a circle because the denominators are equal. If the denominators are different, it's an ellipse like in \( \frac{x^2}{25} + \frac{y^2}{16} = 1 \).

Step 3: Evaluate Each Equation

A. \( \frac{x^{2}}{36}-\frac{y^{2}}{13}=1 \): hyperbola; B. \( \frac{x^{2}}{4}+\frac{y^{2}}{4}=1 \): circle; C. \( \frac{x^{2}}{25}+\frac{y^{2}}{16}=1 \): ellipse; D. \( \frac{y^{2}}{25}-\frac{x^{2}}{39}=1 \): hyperbola; G. \( \frac{x^{2}}{16}-\frac{y^{2}}{65}=1 \): hyperbola; E. \( \frac{x^{2}}{17}+\frac{y^{2}}{81}=1 \): ellipse; F. \( \frac{x^{2}}{36}+\frac{y^{2}}{36}=1 \): circle; H. \( \frac{x^{2}}{144}+\frac{y^{2}}{140}=1 \): ellipse.

Key Concepts

Ellipse

An ellipse is one of the fundamental shapes in the study of conic sections. It’s often described mathematically as the set of all points where the sum of the distances from any point on the ellipse to two fixed points (foci) is constant. When looking at the equation of an ellipse, it is typically written as \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\).
In this form, \(a\) and \(b\) represent the semi-major and semi-minor axes of the ellipse. The axis that is longer determines the orientation of the ellipse — whether it is horizontally or vertically elongated.

• If \(a > b\), it is horizontal.
• If \(a < b\), it is vertical.

To determine if an equation represents an ellipse, check if both terms are added and their coefficients are different. If they are equal, it becomes a special case, which is a circle.

Hyperbola

A hyperbola consists of two separate curves known as branches and is another important conic section. It can be defined as the set of all points where the difference in distances to two fixed points (foci) is constant. The standard form of its equation is \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) or \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\).
The form of the equation dictates the orientation of the hyperbola:

• If \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), it opens horizontally.
• If \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\), it opens vertically.

The hyperbola is characterized by its asymptotes, which are lines that the hyperbola approaches but never touches. These lines intersect at the hyperbola's center and provide guidance on the direction and steepness of the branches.

Circle

A circle is the simplest form and special case of an ellipse where the distances from the center to any point on the shape are equal. Thus, its equation is written as \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), but with a crucial condition where \(a^2 = b^2\).
This condition ensures that the shape is perfectly symmetric regardless of the axis, attributing to its perfect roundness. It means the radius is equal in all directions from the center, making an entity that is rotationally symmetric.
A circle can be easily identified among conic sections equations by verifying if both of the denominators, \(a^2\) and \(b^2\), are the same in the equation that is in the form of an ellipse.

Eccentricity

Eccentricity is a vital measurement in understanding conic sections, as it describes how much a conic section deviates from being circular. It is denoted by the letter \(e\), and its value determines the shape:

• \(e = 0\) indicates a perfect circle.
• \(0 < e < 1\) signifies the shape is an ellipse.
• \(e = 1\) defines a parabola (not discussed here).
• \(e > 1\) describes a hyperbola.

Eccentricity is calculated differently for ellipses and hyperbolas:
For an ellipse, \(e = \frac{c}{d}\) where \(c^2 = |a^2 - b^2|\) and \(d\) is the larger value of \(a\) or \(b\).
For a hyperbola, \(e = \frac{c}{d}\) where \(c^2 = a^2 + b^2\) and \(d\) equals \(a\) or \(b\), depending on the intercepts. Understanding eccentricity is crucial to classifying and predicting orbits such as those of planets and satellites.