Problem 45 - College Algebra with Corequisite Support

Question

For the following exercises, given information about the graph of the hyperbola, find its equation. Vertices at (3,0) and (-3,0) and one focus at (5,0) .

Step-by-Step Solution

Answer

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

Step 1: Understand the Hyperbola Orientation

The vertices are at (3,0) and (-3,0), which means the hyperbola is oriented horizontally along the x-axis. This orientation suggests that the standard form of the equation is \( \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 \).

Step 2: Identify the Center

The center of the hyperbola, \((h, k)\), is the midpoint of the vertices. This means \( h = \frac{3 + (-3)}{2} = 0 \) and \( k = 0 \). Thus, the center is at (0, 0).

Step 3: Determine 'a'

The distance from the center to each vertex is \( a \). Since the vertices are at (3,0) and (-3,0), \( a = 3 \). Therefore, \( a^2 = 9 \).

Step 4: Use Focus to Find 'c'

A focus of the hyperbola is at (5,0). The distance from the center to a focus is \( c \). Therefore, \( c = 5 \), and \( c^2 = 25 \).

Step 5: Calculate 'b^2' Using the Hyperbola Relationship

For hyperbolas, \( c^2 = a^2 + b^2 \). We have \( 25 = 9 + b^2 \). Solving for \( b^2 \), we get \( b^2 = 16 \).

Step 6: Write the Equation of the Hyperbola

Using the values found for \( a^2 \), \( b^2 \), and the center \((h, k)\), the equation is \( \frac{x^2}{9} - \frac{y^2}{16} = 1 \).

Key Concepts

Hyperbola properties

A hyperbola is a fascinating type of conic section defined as the set of points where the difference of the distances to two fixed points (called foci) is constant. Hyperbolas have several important properties that make them stand out in geometry.

Some key characteristics of hyperbolas include:

• **Vertices:** The closest points on the hyperbola to the center. In our example, these points are at (3,0) and (-3,0). These vertices help determine the orientation of the hyperbola.
• **Foci:** These are important points that lie on the axis of symmetry, either inside or outside the hyperbola, and are used to define its shape. For this exercise, one of the foci is at (5,0).
• **Center:** The midpoint between the vertices, which acts as a reference point for the hyperbola's equation. Here, the center is at (0,0).
• **Axes:** The transverse axis runs through the vertices. In this case, it is horizontal, indicating a horizontally oriented hyperbola.

Understanding these properties helps when working with or graphing hyperbolas. They provide a foundation for mastering more complex problems.

Graphing hyperbolas

Graphing a hyperbola can seem challenging at first, but once you understand its elements and structure, it becomes much simpler. The graph of a hyperbola consists of two separate curves that mirror each other. These curves approach asymptotes as they extend outward.

To graph a hyperbola, follow these steps:

• **Identify the Center:** First, locate the center of the hyperbola on the coordinate plane. For our exercise, the center is (0, 0).
• **Plot the Vertices:** Next, mark the vertices along the x-axis at (3,0) and (-3,0). These indicate how wide the hyperbola opens.
• **Asymptotes:** Draw diagonal lines through the center. These lines represent the asymptotes that the hyperbola approaches but never touches. The slopes of these lines depend on the values of 'a' and 'b' (the lengths of the segments from the center to each vertex and a point called the co-vertex, respectively). For this hyperbola, the slopes are determined by the \( \pm \frac{b}{a} \).
• **Sketch the Hyperbola:** With the vertices and asymptotes in place, draw the curves of the hyperbola. They will open outwards along the x-axis, creating two mirror-image curves.

Remember, patience and practice are key when graphing hyperbolas. With time, recognizing their shapes and properties will become second nature.

Conic sections

Conic sections are the curves obtained by intersecting a plane with a cone. Depending on the angle of the intersection and the position of the cone, different types of curves can be created: circles, ellipses, parabolas, and hyperbolas.

Hyperbolas are unique among conic sections:

• **Ellipses** look like stretched circles and are formed when the plane cuts through both nappes of the cone at an angle.
• **Parabolas** are formed when the plane is parallel to the cone's lateral surface.
• **Hyperbolas** are formed when the plane cuts through both nappes, but at such an angle that the intersection creates two separate curves.

The conic section of a hyperbola results in two separated curves that open away from each other, unlike ellipses and parabolas that form closed or singular open shapes. This defining characteristic allows hyperbolas to have their distinct set of applications in fields like astronomy, navigation, and mathematics. Understanding the conic sections helps in visualizing and categorizing these curves effectively.

Standard form hyperbola equation

The standard form of the hyperbola equation provides a clear mathematical representation of the curve. It importantly shows how the curve is positioned on a coordinate plane.

For horizontal hyperbolas, the standard form is:\[ \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 \]

• **(h, k):** Represents the center of the hyperbola. This is the origin in our exercise, i.e., (0, 0).
• **a and b:** These are measurements related to the vertices and the direction in which the hyperbola opens. For our hyperbola, \( a = 3 \), resulting in \( a^2 = 9 \), while \( b^2 = 16 \).

The 'a' term corresponds to the direction of opening along the x-axis for a horizontally oriented hyperbola. The 'b' term, derived from the relationship \( c^2 = a^2 + b^2 \), helps define the asymptotes' slopes.

Using this equation structure, you can easily graph the hyperbola and observe its specific properties, aligning with the mathematical foundation laid out by the standard form.