Problem 46 - College Algebra

Question

Given information about the graph of the hyperbola, find its equation. Vertices at \((0,6)\) and \((0,-6)\) and one focus at \((0,-8) .\)

Step-by-Step Solution

Answer

The equation of the hyperbola is \( \frac{y^2}{36} - \frac{x^2}{28} = 1 \).

Step 1: Understand the Structure of a Hyperbola

The vertices of the hyperbola are given as \((0, 6)\) and \((0, -6)\), which means it is a vertical hyperbola. A vertical hyperbola has an equation of the form \[ \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \] because the transverse axis is vertical.

Step 2: Find the Value of \(a\)

The distance between the vertices is \((6 - (-6)) = 12\). Since this distance is \(2a\), we have \[ 2a = 12, \quad \text{so} \quad a = 6. \]

Step 3: Use the Focus to Find \(c\)

The focus is given as \((0, -8)\), which is \(c\) units from the center at \((0, 0)\). The distance from the center to the focus is \[ c = 8. \]

Step 4: Use the Relationship Between \(a\), \(b\), and \(c\)

For hyperbolas, the relationship is \(c^2 = a^2 + b^2\). We have \(a = 6\) and \(c = 8\). Plug these into the equation to find \(b\):\[ 8^2 = 6^2 + b^2 \]\[ 64 = 36 + b^2 \]\[ b^2 = 28. \]

Step 5: Write the Equation of the Hyperbola

Using the values of \(a^2\) and \(b^2\), the equation of the hyperbola is:\[ \frac{y^2}{36} - \frac{x^2}{28} = 1. \]

Key Concepts

Vertices of Hyperbola

The vertices of a hyperbola are significant points that define its shape and orientation. For any hyperbola, the vertices lie on the transverse axis. In our specific problem, the vertices are at

• \((0, 6)\)
• \((0, -6)\)

These indicate a vertical orientation since their positions share the same x-coordinate. This means that the transverse axis, which is the line segment that passes through the vertices, is vertical.
For hyperbolas, the distance between the vertices is significant as it equals \(2a\), where \(a\) is a parameter that affects the spread of the hyperbola. In our example, this distance is 12, so \(a = 6\). Knowing the vertices allows us to start forming the equation of the hyperbola.

Vertical Hyperbola

When the transverse axis of a hyperbola is vertical, the hyperbola is called a vertical hyperbola. This orientation affects the structure of its equation. A vertical hyperbola has the standard form:

• \[ \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \]

Here, the \(y^2\) term comes first, indicating the vertical transverse axis. A quick way to remember this is "vertical \(\Rightarrow\) \(y^2\) first."
In the vertical hyperbola, the main stretch happens vertically, resulting in vertical opening branches. On a graph, the noticeable characteristic of vertical hyperbolas is how they fan out from the center along the y-axis.

Relationship Between a, b, c in Hyperbolas

For hyperbolas, understanding the relationship between \(a\), \(b\), and \(c\) is crucial in finding its equation. This relationship is defined as:

• \[ c^2 = a^2 + b^2 \]

Here, \(a\), \(b\), and \(c\) represent different elements:

• \(a\): Distance from the center to the vertices along the transverse axis.
• \(b\): A measure that helps define the conjugate axis, impacting the spread horizontally.
• \(c\): Distance from the center to the foci, which are additional key points defining the hyperbola's shape.

Through this exercise, knowing \(a = 6\) and \(c = 8\) allowed us to calculate \(b^2\) using \(a^2 = 36\) and the equation for \(c^2\). It's a mathematical dance between these parameters that perfectly aligns the hyperbola's shape.

Hyperbola with Given Focus

The focal points of a hyperbola, referring specifically to its foci, are another defining characteristic. In our example, one focus is at \((0, -8)\). The foci are located at \((0, c)\) and \((0, -c)\) when centered at the origin.
These points are essential because they determine the hyperbola's eccentricity and directly influence how "wide" or "narrow" it appears. The distance from the center to a focus is \(c\), which we've found to be 8.
Knowledge of the focus helps in verifying our equation and confirms the hyperbola's overall parameters. This information ties back to the formula \(c^2 = a^2 + b^2\), ensuring all parts of the hyperbola's equation work together harmoniously.