Problem 25 - Precalculus with Limits

Question

In Exercises 23-28, find the center, vertices, foci, and the equations of the asymptotes of the hyperbola. Use a graphing utility to graph the hyperbola and its asymptotes. \(4x^2-9y^2=36\)

Step-by-Step Solution

Answer

The center of the given hyperbola is at (0,0), the vertices are at (±3,0), the foci are at (±\sqrt{13}, 0) and the equations of the asymptotes are \(y=\pm \frac{2}{3}x\).

Step 1: Rewrite the formula

The given equation is \(4x^2-9y^2=36\). By dividing all terms by 36 to get the equation into the form \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), where \(a\) and \(b\) are the lengths of the semi-axes, we get \(\frac{x^2}{9} - \frac{y^2}{4} = 1\)

Step 2: Identify the center, vertices, and foci

The center of the hyperbola is at (0,0) because there are no shifts in the equation. The vertices are at (±a, 0), so in this case they are (±3,0). For the foci, we need to find 'c', where \(c^2 = a^2 + b^2\). Substituting, \(c^2 = 9+4 =13\), so \(c = \sqrt{13}\), and the foci are located at (±c, 0) = (±\sqrt{13}, 0)

Step 3: Find the equations of the asymptotes

The equations of the asymptotes for a horizontal hyperbola are \( y=\pm \frac{b}{a}x\). Substituting values of 'a' and 'b', the equations of the asymptotes are \(y=\pm \frac{2}{3}x\)

Key Concepts

Conic Sections

Conic sections are shapes created as a plane intersects with a cone. Depending on the angle and position of the intersection, the shape can be a circle, ellipse, parabola, or a hyperbola. The unique characteristics of each shape are determined by the relative position of the intersecting plane to the cone's base and side.

In the case of a hyperbola, the plane cuts through both nappes (the upper and lower cones) but does not intersect the apex of the cones. This results in two mirror-image curves that are open in opposite directions. Each of these curves is one branch of the hyperbola, and they are separated by an asymptotic gap which the curves approach but never cross. For students, it's crucial to visualize this as it lays the foundation for understanding hyperbolas and their equations.

Hyperbola Equations

The standard equation of a hyperbola with horizontal transverse axis is \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\] and the one with vertical transverse axis is \[\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\]. In these equations, \(a\) and \(b\) represent the distances from the center to the vertices and to the ends of the conjugate axis, respectively.

To graph a hyperbola from its equation, it is essential to rewrite the equation so that one side equals one, which is essential for identifying the distances \(a\) and \(b\). Then, by locating the center and the axes distances, students can sketch the shape of the hyperbola, before refining it by plotting additional points and determining the direction in which the branches open.

Hyperbola Characteristics

Hyperbolas possess distinct characteristics that set them apart from other conic sections:

• They consist of two separate curves called branches.
• The center of a hyperbola is the midpoint between its two foci.
• Hyperbolas have two axes of symmetry - the transverse axis which contains the vertices, and the conjugate axis which is perpendicular to it at the center.
• Each branch of a hyperbola asymptotically approaches two lines called asymptotes, which define the slant of the branches but are never touched or crossed by the hyperbola itself.

Understanding these characteristics helps in analyzing the hyperbola equation and in graphing the conic accurately. The vertices and foci are particularly significant because they determine the shape and orientation of the hyperbola.

Asymptotes of Hyperbola

Asymptotes are lines that a curve approaches infinitely closely but never actually meets. For hyperbolas, asymptotes are crucial for sketching the graph as they provide a skeleton which guides the shape of the hyperbola's branches.

The asymptotes of a hyperbola with horizontal transverse axis can be found with the formula \[y = \pm \frac{b}{a}x\], where \(a\) and \(b\) are the lengths of the semi-axes from the standard hyperbola equation. These lines pass through the center of the hyperbola and their slopes determine the direction in which the branches open.

It is beneficial for students to recognize that these asymptotes are the diagonals of a rectangle centered at the hyperbola's center and having sides of lengths 2a and 2b. This rectangle helps in establishing a boundary that the hyperbola will never exceed, simplifying the graphing process.