Problem 37 - Thomas Calculus

Question

Exercises \(35-38\) give information about the foci, vertices, and asymptotes of hyperbolas centered at the origin of the \(x y\) -plane. In each case, find the hyperbola's standard-form equation from the information given. $$ \begin{array}{l}{\text { Vertices: }( \pm 3,0)} \\ {\text { Asymptotes: } y=\pm \frac{4}{3} x}\end{array} $$

Step-by-Step Solution

Answer

The equation of the hyperbola is \(\frac{x^2}{9} - \frac{y^2}{16} = 1\).

Step 1: Identify Hyperbola Orientation

Since the vertices are located at \((\pm 3, 0)\), the hyperbola is centered at the origin and opens horizontally. For a horizontally opening hyperbola, the standard form of the equation is \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\).

Step 2: Find the Value of 'a'

The distance from the center to each vertex along the x-axis is the value of \(a\). Given that the vertices are \((\pm 3, 0)\), we have \(a = 3\). Thus, \(a^2 = 9\).

Step 3: Use Asymptotes to Find 'b'

The slopes of the asymptotes \(y = \pm \frac{4}{3} x\) tell us that \(\frac{b}{a} = \frac{4}{3}\). Using \(a = 3\), we compute \(b = \frac{4}{3} \times 3 = 4\). Thus, \(b^2 = 16\).

Step 4: Write the Equation of the Hyperbola

With \(a^2 = 9\) and \(b^2 = 16\), we write the standard form of the hyperbola: \(\frac{x^2}{9} - \frac{y^2}{16} = 1\).

Key Concepts

Standard-Form Equation

The standard-form equation of a hyperbola helps to describe its important features and general orientation in the Cartesian plane. The equation for a horizontally oriented hyperbola centered at the origin is \[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \].
This equation consists of two squared terms - one for the x-direction and one for the y-direction. The key to recognizing a hyperbola is the subtraction between these two terms. Here, \( a^2 \) represents the square of the distance from the center to the vertex along the x-axis, while \( b^2 \) comes from the asymptote's slope relationship.
Using given information like vertices and asymptotes, we can deduce exact values for \( a^2 \) and \( b^2 \) to fully articulate the hyperbola's equation.

Vertices

Vertices of a hyperbola are the points where the hyperbola intersects its axis. They are important as they indicate the "widest" points of the hyperbola.
For a hyperbola centered at the origin and horizontally oriented, the vertices have the coordinates \((\pm a, 0) \). In the given exercise, the vertices are \((\pm 3, 0) \), which suggests \( a = 3 \). Since the vertices lie along the x-axis, this tells us about the horizontal nature of the hyperbola.

• The distance from the center to a vertex is exactly \( a \).
• For problems like these, identifying the vertex positions is a straightforward way to determine \(a\) and calculate \(a^2\).

Therefore, \( a^2 = 9 \), aligning with the hyperbola's standard-form equation.

Asymptotes

Asymptotes of a hyperbola are crucial because they indicate directions that the hyperbola approaches but never touches. In this exercise, the asymptotes are given as \( y = \pm \frac{4}{3} x \), suggesting that the slope of the asymptote lines is \( \frac{b}{a} = \frac{4}{3} \).
To find \( b \), we need to use this slope relationship:

• We already have \( a = 3 \), so we use \( \frac{b}{3} = \frac{4}{3} \).
• This equation tells us \( b = 4 \).
• Squaring \( b \), we find \( b^2 = 16 \).

These values for \( b^2 \) and \( a^2 \) allow us to finalize the hyperbola's standard-form equation. Understanding the role of asymptotes can greatly simplify solving hyperbola-related problems.

Mathematical Problem Solving

Navigating through a mathematical problem like determining a hyperbola's equation involves strategic use of given information. Here's a general approach:

• Identify what information you have—such as vertices and asymptotes.
• Choose the appropriate form of the equation based on the orientation of the hyperbola.
• Plug in provided measurements step by step—first, find \( a \) from the vertices, then \( b \) using the slope of the asymptotes.
• Verify by substituting values back into the equation.

By methodically working through each step, complex problems become more manageable. Each piece of information helps to construct the final equation and solve the problem accurately. Practice solid mathematical problem-solving skills by understanding the role each component plays, which leads you to complete and precise solutions.