Problem 48 - College Algebra

Question

For the following exercises, given information about the graph of the hyperbola, find its equation. Center: (0,0) ; vertex: (0,-13) ; one focus: \((0, \sqrt{313})\).

Step-by-Step Solution

Answer

The equation of the hyperbola is \( \frac{y^2}{169} - \frac{x^2}{144} = 1 \).

Step 1: Identifying the Orientation of the Hyperbola

Since the vertex and the focus both lie on the y-axis, the hyperbola is a vertical hyperbola. The standard form of the equation of a vertical hyperbola centered at the origin is \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \).

Step 2: Determining 'a' from the Vertex

The distance from the center to the vertex is \( a \). Since the vertex is at (0,-13), we have \( a = 13 \). Thus, \( a^2 = 169 \).

Step 3: Determining 'c' from the Focus

The distance from the center to the focus is \( c \). With the given focus at \( (0, \sqrt{313}) \), we have \( c = \sqrt{313} \). Thus, \( c^2 = 313 \).

Step 4: Using the Relationship Between 'a', 'b', and 'c'

For hyperbolas, the relationship between \( a \), \( b \), and \( c \) is \( c^2 = a^2 + b^2 \). Substitute the known values: \( 313 = 169 + b^2 \).

Step 5: Solving for 'b^2'

Rearrange and solve for \( b^2 \): \( b^2 = 313 - 169 = 144 \).

Step 6: Writing the Equation of the Hyperbola

Now that we have \( a^2 = 169 \) and \( b^2 = 144 \), substitute these into the standard form: \( \frac{y^2}{169} - \frac{x^2}{144} = 1 \).

Key Concepts

Standard Form of Hyperbola

When learning about hyperbolas, one of the most fundamental aspects is their equation, which comes in a specific "standard form." This form is essential because it helps us understand and graph the hyperbola's characteristics. The standard form of a hyperbola centered at the origin is generally given in two ways, depending on whether it is horizontal or vertical:

For horizontal hyperbolas, the equation is:

• \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \)

For vertical hyperbolas, the equation is:

• \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \)

Both forms showcase the variables \( a^2 \) and \( b^2 \), which relate to the distances to the vertices and other critical points. In essence, \( a \) represents the distance from the center to the vertices along the main axis of the hyperbola, and \( b \) corresponds to the distance along the other axis that passes through the center.

The terms \( a^2 \) and \( b^2 \) help define the shape and size of the hyperbola, making it crucial to determine them exactly when finding the hyperbola's equation.

Vertical Hyperbola

A vertical hyperbola orients itself in a way that makes its main or transverse axis parallel to the y-axis. When the problem states that both the vertex and one focus lie on the y-axis, it confirms that we are dealing with a vertical hyperbola.

Key characteristics of a vertical hyperbola include:

• The equation takes the form \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \), where \( a^2 \) is associated with the y-component, emphasizing the vertical orientation.
• Its center remains at the origin, \( (0,0) \), unless specified otherwise.
• The vertices lie at \( (0,\pm a) \) along the y-axis, while foci lie further out at \( (0, \pm c) \), with \( c > a \).

Visualizing a vertical hyperbola helps conceptualize how curves extend outwardly from the center following these conventions. Recognizing the orientation guides the setup and solving of the equation based on how it covers vertical distances pertaining to \( a \) and the focus determined by \( c \).

This orientation insight is not just numerical but visual when graphing or interpreting data, as it significantly affects how the hyperbola appears on a coordinate plane.

Relationship Between a, b, and c in Hyperbolas

Understanding the relationship between \( a \), \( b \), and \( c \) in hyperbolas is crucial as it directly affects the equation of the curve. In hyperbolas, the fundamental relationship is governed by the equation:

• \( c^2 = a^2 + b^2 \)

This equation helps calculate unknown variables if one or more are known. Here's what each variable signifies:

• \( a \) is the distance from the center to each vertex, forcefully defining the reaching extent of the hyperbola along its central axis.
• \( c \) is the distance from the center to each focus, which are points located further from the center than the vertices.
• \( b \) is associated more subtly with the "box" that guides asymptotes, often less visible but equally important in how the hyperbola expands.

By effectively understanding this relationship, one can deduce any missing dimension \( a \), \( b \), or \( c \) using the information provided about the hyperbola. It's an essential aspect when solving problems or deriving equations for hyperbolas based on their geometric properties and points. Identifying this relationship not only solves the equations but adds clarity to their structural nature.