Problem 68 - Intermediate Algebra

Question

Compare the graphs of \(\frac{x^{2}}{81}-\frac{y^{2}}{64}=1\) and \(\frac{y^{2}}{81}-\frac{x^{2}}{64}=1 .\) Do they have any similarities?

Step-by-Step Solution

Answer

The hyperbolas are similar; they have the same axes lengths and differ only in orientation, being rotated 90 degrees from each other.

Step 1: Identify the Graphs

The equations given are both equations of hyperbolas. The standard form for a hyperbola centered at the origin with a horizontal transverse axis is \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), while for a vertical transverse axis it's \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \).

Step 2: Analyze the First Equation

The equation \( \frac{x^{2}}{81} - \frac{y^{2}}{64} = 1 \) is a hyperbola with a horizontal transverse axis. Here, \( a^2 = 81 \) and \( b^2 = 64 \), so \( a = 9 \) and \( b = 8 \).

Step 3: Analyze the Second Equation

The equation \( \frac{y^{2}}{81} - \frac{x^{2}}{64} = 1 \) is a hyperbola with a vertical transverse axis. In this case, \( a^2 = 81 \) and \( b^2 = 64 \), thus \( a = 9 \) and \( b = 8 \) as well.

Step 4: Determine the Vertices and Asymptotes

For the first hyperbola, the vertices are at \((\pm9, 0)\) and the asymptotes are \(y = \pm\frac{8}{9}x\). For the second hyperbola, the vertices are at \((0, \pm9)\) and the asymptotes are \(y = \pm\frac{9}{8}x\).

Step 5: Compare the Similarities

Both hyperbolas have the same semi-major and semi-minor axes lengths (9 and 8 respectively), which means they open in perpendicular directions but have the same shape. The lengths of the axes and the distance of the vertices to the center are identical, only rotated 90 degrees around the center.

Key Concepts

Horizontal Transverse Axis

In a hyperbola, the transverse axis determines the direction in which the hyperbola opens. For the equation of the form \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), the transverse axis is horizontal. This means that the hyperbola opens left and right along the x-axis.

Key features for a hyperbola with a horizontal transverse axis include:

• Vertices located at \((\pm a, 0)\). These are the points where the hyperbola intersects the transverse axis.
• The length of the transverse axis is \(2a\), which represents the distance between the vertices across the x-axis.
• The foci, which are points inside each branch of the hyperbola, are located at \((\pm c, 0)\), where \(c = \sqrt{a^2 + b^2}\).

Understanding the direction of a hyperbola’s transverse axis is crucial for sketching its graph.

Vertical Transverse Axis

When dealing with hyperbolas, the orientation of the transverse axis directly affects its shape and appearance. For a hyperbola expressed by the equation \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \), the transverse axis is vertical.

This means the hyperbola opens upward and downward along the y-axis. Important features include:

• Vertices located at \((0, \pm a)\), indicating where the hyperbola intersects the vertical transverse axis.
• The length of the transverse axis, \(2a\), similarly determines the spread between the vertices upwards and downwards.
• The foci follow the same logic and are positioned at \((0, \pm c)\), with \(c = \sqrt{a^2 + b^2}\).

This configuration shifts the orientation entirely, giving it a distinct visual form compared to its horizontal counterpart.

Vertices

For hyperbolas, the vertices are key points where the hyperbola meets its transverse axis. They serve as the turning points, marking the center of each of the hyperbola’s branches.

In the context of the standard hyperbola equations:

• For a horizontal transverse axis, as in \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), the vertices are at \((\pm a, 0)\). These represent the widest points along the x-axis.
• For a vertical transverse axis, with the form \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \), the vertices are at \((0, \pm a)\). Here, they mark the maximum height above and below the center.

Considering the vertices helps determine the orientation and width of the hyperbola.

Asymptotes

Asymptotes in hyperbolas are crucial lines that fundamentally affect the shape and direction of a hyperbola’s branches. For any hyperbola, asymptotes intersect at the center, shaping an "X" that the hyperbola approaches but never quite touches.

Depending on the equation of the hyperbola:

• With a horizontal transverse axis, as in \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), the asymptotes have the equations \( y = \pm \frac{b}{a} x \), forming a cross through the origin with gentle slopes.
• When the transverse axis is vertical, like in \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \), the asymptotes are given by \( y = \pm \frac{a}{b} x \). The slopes are steeper, guiding the hyperbola's vertical stretch.

These asymptotes play a critical role in determining the hyperbola’s overall form and are essential in sketching an accurate graph.