Problem 69 - College Algebra

Question

Describe how to locate the foci of the graph of \(\frac{x^{2}}{9}-\frac{y^{2}}{1}=1\) Describe one similarity and one difference between the graphs of \(\frac{x^{2}}{9}-\frac{y^{2}}{1}=1\) and \(\frac{y^{2}}{9}-\frac{x^{2}}{1}=1\)

Step-by-Step Solution

Answer

The foci of the given hyperbola \(\frac{x^{2}}{9}-\frac{y^{2}}{1}=1\) are located at (\(\pm \sqrt{10},0\)). The two hyperbolas are similar in that they both have the same absolute values for a and b, that determine the x and y intercepts, but they differ in the way they are oriented. The first hyperbola is horizontal, while the second one is vertical.

Step 1: Identifying the hyperbola

Given \(\frac{x^{2}}{9}-\frac{y^{2}}{1}=1\), it's clear that the equation is that of a hyperbola since one term is subtracted from the other. Here the x-term is positive so it's a horizontal hyperbola.

Step 2: Locating the foci of the hyperbola

For a hyperbola, the distance from the center to each focus is given by \(c=\sqrt{a^{2}+b^{2}}\). Here, \(a^2 = 9\) and \(b^2=1\), so \(c = \sqrt{9+1} = \sqrt{10}\). Hence the foci of the hyperbola \(\frac{x^{2}}{9}-\frac{y^{2}}{1}=1\) are located at (\(\pm \sqrt{10},0\)).

Step 3: Comparing the two hyperbolas

The equation \(\frac{y^{2}}{9}-\frac{x^{2}}{1}=1\) is also a hyperbola, but with the y-term being positive, hence it's a vertical hyperbola. The similarity lies in the fact that both are hyperbolas and have the same absolute values for a and b (which determine the x and y intercepts respectively). The difference is simply the orientation of these hyperbolas - the first hyperbola is oriented horizontally while the second is oriented vertically.

Key Concepts

Understanding Foci of a Hyperbola

The foci (plural of focus) are special points located along the axis of a hyperbola. These points play a critical role in defining the shape and properties of the hyperbola itself. In mathematics, for a given hyperbola, the distance from the center to each focus is calculated using the formula: \[ c = \sqrt{a^{2} + b^{2}} \]

• The variable \(a\) is associated with the squared term of the variable that has a positive coefficient.
• The variable \(b\) is associated with the squared term of the variable that has a negative coefficient.

For example, in the hyperbola equation \( \frac{x^{2}}{9} - \frac{y^{2}}{1} = 1 \), we have \( a^{2} = 9 \) and \( b^{2} = 1 \), leading to:\[ c = \sqrt{9 + 1} = \sqrt{10} \]So, the foci are located at points \( (\pm \sqrt{10}, 0) \) along the x-axis. Each focus aids in creating the distinctive shape of the hyperbola. Both foci are equidistant from the center of the hyperbola.

Defining a Horizontal Hyperbola

A horizontal hyperbola is characterized by the placement of its transverse axis along the x-axis. In technical terms, if the x-term is positive in the hyperbola's equation, the hyperbola opens horizontally.
Consider the equation: \( \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 \). Here:

• The positive \(x^2\) term indicates a horizontal orientation.
• The foci are positioned on a horizontal line parallel to the x-axis at \( \pm c \).

For the given hyperbola \( \frac{x^{2}}{9} - \frac{y^{2}}{1} = 1 \), the structure is such that the vertices and foci align horizontally along the x-axis. This means that both the shape and orientation are dictated by how the equation is structured, placing emphasis on the x component.

Recognizing a Vertical Hyperbola

A vertical hyperbola has its transverse axis positioned along the y-axis, distinguishing it from the horizontal orientation. If the y-term is positive in the hyperbola's equation, the hyperbola is vertical.
Consider the equation: \( \frac{y^{2}}{a^{2}} - \frac{x^{2}}{b^{2}} = 1 \). Here:

• The positive \(y^2\) term dictates a vertical orientation.
• The vertices and foci are located along the vertical y-axis, positioned vertically at \( \pm c \).

For the example equation \( \frac{y^{2}}{9} - \frac{x^{2}}{1} = 1 \), we see that the arrangement causes the hyperbola to extend vertically. This makes the shape appear taller, with the major axis aligned with the y-axis, rather than the x-axis as seen in horizontal hyperbolas.

Exploring the Distance Formula in Hyperbolas

The distance formula is essential to finding the positions of the foci in a hyperbola. It provides a method to calculate the exact distance from the center to each focus, ensuring the correct geometry is maintained. The formula used is: \[ c = \sqrt{a^{2} + b^{2}} \]

• \(a\) and \(b\) are derived from the denominators in the hyperbola equation \( \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 \).
• \(a^{2} = 9\) represents the larger axis, while \(b^{2} = 1\) relates to the minor axis.

Understanding how to apply and manipulate the distance formula is key to graphing hyperbolas accurately. This can make it easier to determine where the foci are situated, which in turn affects how we visualize and interpret the hyperbola in a coordinate plane.