Problem 22 - Precalculus: Functions and Graphs

Question

Find an equation for the hyperbola that has its center at the origin and satisfies the given conditions. Foci \(F(\pm 8,0)\) vertices \(V \pm 5,0)\)

Step-by-Step Solution

Answer

The hyperbola equation is \( \frac{x^2}{25} - \frac{y^2}{39} = 1 \).

Step 1: Understand hyperbola standard form

The standard form of a hyperbola centered at the origin is \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) for horizontal hyperbolas. Here, \(a\) is the distance from the center to each vertex, and \(c\) is the distance from the center to each focus. We need to find \(a\) and \(b\).

Step 2: Identify given values

From the problem, the vertices \( V(\pm 5, 0) \) give us \( a = 5 \). The foci \( F(\pm 8, 0) \) give us \( c = 8 \).

Step 3: Use the hyperbola relationship

For a hyperbola, the relationship between \(a\), \(b\), and \(c\) is \( c^2 = a^2 + b^2 \). Using the given information, we have \( c^2 = 8^2 = 64 \) and \( a^2 = 5^2 = 25 \).

Step 4: Solve for \(b^2\)

Using the equation from Step 3, substitute the known values: \( 64 = 25 + b^2 \). Solving for \(b^2\), we find \( b^2 = 64 - 25 = 39 \).

Step 5: Write the equation of the hyperbola

Substitute \(a^2=25\) and \(b^2=39\) into the standard form equation \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \). Thus, the equation of the hyperbola is \( \frac{x^2}{25} - \frac{y^2}{39} = 1 \).

Key Concepts

Center at the Origin

The center of a hyperbola is the point around which the hyperbola is balanced. In mathematical terms, it's the midpoint between the two foci of the hyperbola. When we say the center is 'at the origin,' we mean

• The center is located at the coordinate point \((0,0)\).
• The hyperbola is symmetrically placed around this point in the coordinate plane.

The significance of having the center at the origin is that it simplifies the equation of the hyperbola greatly. It makes our calculations straightforward, as we don't have to adjust for any shifts in the coordinate axes. Instead, all the values we calculate can be easily substituted into the standard hyperbola equation without additional transformations.

Foci of Hyperbola

The foci (pronounced 'foh-sigh') are specific points inside a hyperbola. These points are vital in defining the shape and orientation of the hyperbola. For a hyperbola with a center at the origin, foci are usually given as \((\pm c, 0)\) for a horizontally oriented hyperbola. The distance from the center to each focus is denoted by the letter \(c\). In our example, the foci are located at \((\pm 8, 0)\), which tells us that \(c = 8\).

• The value of \(c\) determines how narrowly or widely the hyperbola opens along its major axis.
• Calculation of \(c\) is crucial to determine \(b\), another key component of the hyperbola's equation.

Vertices of Hyperbola

The vertices of a hyperbola can be thought of as the 'corner' points where each branch of the hyperbola is the closest to the center. If the hyperbola is horizontally oriented, these vertices are located along the x-axis. For our hyperbola, the vertices are given as \((\pm 5, 0)\). This implies that the distance \(a = 5\), where \(a\) is the distance from the center to each vertex.

• Vertices help in identifying the major axis of the hyperbola. For horizontal hyperbolas, the major axis lies along the x-axis.
• Knowing the vertices aids in determining the value of \(a\), which is used in the standard form of the hyperbola equation.

Standard Form of Hyperbola

A hyperbola in standard form indicates its orientation and the relationship between its major and minor axes. For a hyperbola centered at the origin and opening horizontally, the standard form is written as \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \).

• The term \(\frac{x^2}{a^2}\) depicts the major axis alignment along the x-axis, and \(a\) is the distance from the center to each vertex.
• The term \(\frac{y^2}{b^2}\) involves \(b\), representing the relationship with the other axis (y-axis in this case).
• In our particular case, using \(a=5\) and \(b=\sqrt{39}\) give us the complete equation of the hyperbola as \(\frac{x^2}{25} - \frac{y^2}{39} = 1\). This equation illustrates both the hyperbola’s shape and size based on the standard form requirements.