Problem 80 - Precalculus with Limits

Question

CAPSTONE Given the hyperbolas \(\dfrac{x^2}{16}-\dfrac{y^2}{9}=1 \quad\) and \(\quad \dfrac{y^2}{9}-\dfrac{x^2}{16}=1\) describe any common characteristics that the hyperbolas share, as well as any differences in the graphs of the hyperbolas. Verify your results by using a graphing utility to graph each of the hyperbolas in the same viewing window.

Step-by-Step Solution

Answer

The given are a pair of conjugate hyperbolas, sharing the same center, the same distances from the center to each vertex and each focus, yet one opens horizontally (left-right) and the other vertically (up-down).

Step 1: Standard Form of a Hyperbola

The standard form of a hyperbola centered at the origin is \((x^2/a^2) - (y^2/b^2) = 1\) for a hyperbola opening left and right, and \((y^2/a^2) - (x^2/b^2) = 1\) for a hyperbola opening up and down. The larger denominator determines the orientation.

Step 2: Inspect the Given Hyperbolas

Given hyperbolas are \((x^2/16) - (y^2/9) = 1\) and \((y^2/9) - (x^2/16) = 1\). The first hyperbola is in the form \((x^2/a^2) - (y^2/b^2) = 1\) where \(a^2 = 16\) and \(b^2 = 9\), so it is horizontal. Similarly, the second hyperbola is vertical because it is in the other standard form with same \(a^2 = 9\) and \(b^2 = 16\).

Step 3: Common Characteristics

Both hyperbolas share the same center (0,0), the same distances from the center to each vertex and each focus. The vertices for each hyperbola are \(a\) units from the center, and the foci are \(\sqrt{a^2 + b^2}\) units from the center.

Step 4: Differences

Although they share common characteristics, one hyperbola opens horizontally (left-right) whereas the other opens vertically (up-down). Their axes are thus perpendicular to each other.

Step 5: Graphical Verification

Use a graphing utility to sketch both hyperbolas. The results will confirm the previous observations, that is, the hyperbolas share the same center, vertices, and foci but open in opposite directions along perpendicular axes.

Key Concepts

Conic Sections

In mathematics, conic sections are curves obtained by intersecting a plane with a double-napped cone. They are dubbed 'conic' because of their geometric origin from a cone. These include circles, ellipses, parabolas, and hyperbolas. Each takes a distinct shape based on the angle and position of the intersection.

Hyperbolas are one of these intriguing conic sections. Unlike circles and ellipses, which are closed curves, hyperbolas are open, appearing as two separate but mirror-image arcs. Their unique properties are gained from the plane slicing the cone in a manner that cuts through both nappes.

Understanding conic sections is essential because they model lots of natural phenomena and structures, from the design of satellite dishes to the path paths of comets around the sun.

Graphing Utilities

Graphing utilities are powerful tools that help individuals visualize mathematical equations and functions effortlessly. They are especially beneficial when dealing with equations like those of conic sections, which can have multiple forms and orientations.

• These utilities enable one to input equations and adjust their viewing window to get a clearer picture of the graph.
• Most come with various features like plotting points, curves, and adjusting parameters to better understand the behavior of a graph.
• They serve as a confirmation tool. For instance, once you've anticipated the shape and orientation of a hyperbola, graphing utilities can visually verify these predictions.

Consequently, they pave the way for deeper understanding and exploration beyond manual plotting, offering vivid insights into the intersection and behavior of conic sections.

Standard Form of Hyperbolas

The standard form of a hyperbola is key to understanding its orientation and structure. There are two primary standard forms for hyperbolas, each associated with a different orientation.

• The form \((x^2/a^2) - (y^2/b^2) = 1\) indicates a hyperbola that opens left and right, horizontally.
• In contrast, the form \((y^2/a^2) - (x^2/b^2) = 1\) reveals a hyperbola that opens vertically, up and down.

In these equations, \(a^2\) and \(b^2\) represent the square of the distances from the center to the vertices and co-vertices, respectively. The major difference between these forms boils down to which variable term is positive.

With the center at the origin \((0,0)\), using the standard form allows for quick deductions about the hyperbola's orientation and crucial attributes like its vertices and foci, helping to graph and understand these shapes more effectively.