Problem 64 - 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 \(A-H,\) find the value of \(d\).

Step-by-Step Solution

Answer

Calculate "d" for each equation by identifying "a" and "b" and applying the rules based on the conic section type.

Step 1: Identify Conic Sections

Examine each equation to determine if it represents a circle, ellipse, or hyperbola. The equations with the sum of squares are ellipses or circles, while those with a difference represent hyperbolas.

Step 2: Determine a and b

For each conic section, express the equation in the form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) for ellipses/circles or \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) or \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\) for hyperbolas. Identify \(a^2\) and \(b^2\) and then take the square root to get \(a\) and \(b\).

Step 3: Calculate "d" for Each Equation

For each ellipse or circle, \(d\) is the greater of \(a\) or \(b\). For each hyperbola in the form \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), \(d = a\). If it's in the form \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\), \(d = b\).

Key Concepts

Eccentricity

In the study of conic sections, eccentricity is a key concept that describes how much a conic deviates from being circular. The eccentricity, denoted as \( e \), is a non-negative number. Each type of conic section has a distinctive range of eccentricities:

• A circle has an eccentricity of 0, indicating it is perfectly round.
• An ellipse has an eccentricity greater than 0 and less than 1, showing varying degrees of flattening.
• A parabola has an eccentricity exactly equal to 1, reflecting its unique open curve.
• A hyperbola has an eccentricity greater than 1, indicating an open form with two branches.

Eccentricity can be calculated using the formula \(e = \frac{c}{d}\), where \(c\) is the distance from the center to the focus, and \(d\) is the semi-major axis for ellipses and the distance to a vertex for hyperbolas. Understanding the eccentricity helps astronomers and mathematicians to determine the nature and shape of orbits and curves.

Ellipses

Ellipses are an interesting type of conic section, characterized by their oval shape. An ellipse is defined as the set of all points such that the sum of the distances from two fixed points, called foci, is constant. In centered ordinary situations, ellipses have equations of the form \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \).
If \(a = b\), the ellipse is actually a circle. However, when \(a eq b\), the ellipse is stretched in the direction of the larger axis. The largest of \(a\) or \(b\) is identified as the semi-major axis. This plays a major role in calculating eccentricity and understanding the stretch of the ellipse. Each ellipse has two axes – the longer major axis and the shorter minor axis – which determine its overall shape.
Ellipses are crucial in various fields, including astronomy, as they describe the orbits of planets and satellites.

Hyperbolas

Hyperbolas are formed by the intersection of a plane with both nappes of a double cone. They consist of two separate curves known as branches. In terms of geometry, a hyperbola is defined as the set of points where the difference in distance to two fixed points (foci) is constant. Its standard form is either \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) or \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \).
For hyperbolas, \(a\) is always associated with the first term in the equation. The line through the centers of both branches is called the transverse axis. The eccentricity of hyperbolas (\(e > 1\)) reflects their two-branched nature and open form. In hyperbolas, instead of a continuous curve like ellipses, you get two diverging paths, making them unique among conic sections.
Understanding hyperbolas aids in areas ranging from navigation systems to the design of certain telescopes.

Standard Form of Conics

The conic sections each have their distinct standard forms which notably help in identifying their type and key properties. Recognizing these standard forms is essential when dealing with equations of conics:

• For circles, the standard form is \(x^2 + y^2 = r^2\), where \(r\) is the radius.
• Ellipses are given by \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), where \(a\) and \(b\) determine the axes lengths.
• Hyperbolas have the form \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) or \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\), which describes their diverging paths.

These forms allow us to easily categorize sections of conics and comprehend their properties by inspecting the coefficients and variable terms. Deriving their standard forms simplifies the process of analyzing their shape and orientation in geometry.

Circle

The circle is the simplest form of conic sections, defined as the set of all points equidistant from a fixed point, known as the center. This equidistance is the radius \(r\), and its equation is \(x^2 + y^2 = r^2\). A circle is essentially a special case of an ellipse, where the lengths of both the semi-major and semi-minor axes are the same.
Circles have an eccentricity of \(0\), reflecting their perfect symmetry and uniformity. They serve as a fundamental shape in geometry and are ubiquitous in the natural and manufactured world. By understanding a circle’s properties and characteristics, one can better appreciate the more complex properties of other conic sections such as ellipses and hyperbolas.
Circles are extensively used in design and engineering because their geometry is fundamental and inherently stable.