Problem 46 - College Algebra

Question

For the following exercises, 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: Identify the Center

The hyperbola is given to have vertices at \((0,6)\) and \((0,-6)\). These points are vertically aligned along the y-axis, suggesting a vertical hyperbola. The center of the hyperbola is the midpoint of the vertices. Thus, the center is located at \((0,0)\).

Step 2: Determine 'a' from Vertices

Since the vertices are at \((0,6)\) and \((0,-6)\), the distance from the center \((0,0)\) to each vertex is 6. This means that \(a = 6\).

Step 3: Determine 'c' from Focus

The hyperbola has a focus at \((0,-8)\). The distance from the center \((0,0)\) to this focus is 8, giving \(c = 8\).

Step 4: Calculate 'b' Using the Relationship 'c^2 = a^2 + b^2'

For hyperbolas, the relation between \(a\), \(b\), and \(c\) is given by \(c^2 = a^2 + b^2\). Substituting the known values: \(8^2 = 6^2 + b^2\), which simplifies to \(64 = 36 + b^2\). Solve for \(b^2\) to get \(b^2 = 28\).

Step 5: Write the Equation of the Hyperbola

For a vertical hyperbola centered at \((0,0)\), the equation is \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\). We substitute \(a^2 = 36\) and \(b^2 = 28\) to get \(\frac{y^2}{36} - \frac{x^2}{28} = 1\). Therefore, the equation of the hyperbola is \(\frac{y^2}{36} - \frac{x^2}{28} = 1\).

Key Concepts

Equation of Hyperbola

A hyperbola is a type of conic section defined by a specific type of equation. To form this equation, the relationship between the variables must meet certain criteria. For a vertical hyperbola centered at the origin \((0,0)\), the general form of the equation is: \[\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\] The terms \(a^2\) and \(b^2\) represent the squares of the distances from the center to each vertex and to the endpoints of the conjugate axis, respectively. It is important to understand that:

• \(a\) is always associated with the vertices.
• \(b\) represents the perpendicular distance to the vertices, known as the conjugate axis.

Hyperbolas have distinctive properties and patterns that help differentiate them from ellipses, other types of conic sections. Keeping track of \(a\), \(b\), and \(c\) (where \(c\) is the distance to the foci), is essential in forming the correct equation for any hyperbola.

Vertices and Foci

Vertices and foci are crucial elements in understanding a hyperbola. The vertices of a hyperbola are the points where the hyperbola intersects its major axis. In our exercise, the vertices at \((0,6)\) and \((0,-6)\) imply that \(a = 6\), as the distance from the center to each vertex is 6 units.The distance \(a\) is pivotal in the equation of a hyperbola.

Foci, on the other hand, are additional points along the same axis as the vertices. They are located further away from the center than the vertices. The given focus at \((0,-8)\) indicates that \(c = 8\).

To understand the relationship between the vertices and foci, remember the equation \(c^2 = a^2 + b^2\). In this case, it helped us find \(b^2 = 28\).These relationships and distances are unique to hyperbolas and aid in transitioning from geometric concepts to algebraic equations.

Center of Hyperbola

The center of a hyperbola is a crucial reference point from which all other key elements are measured. For hyperbolas that are symmetric about the y-axis with vertices and foci distributed evenly, as in our exercise, the center can be found by taking the midpoint of the segment between the vertices.

In this example, the vertices are \((0,6)\) and \((0,-6)\), so the center is located at the midpoint, which is \((0,0)\).The center is often denoted as \((h,k)\), where \(h\) and \(k\) are the x and y coordinates respectively, in the general form of the hyperbola equation \(\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1\). Understanding the location of the center helps in identifying how the hyperbola's branches are aligned in the coordinate plane, and whether it opens upward, downward, or sideways.

Conic Sections

Conic sections arise from the intersection of a plane with a double-napped cone and include circles, ellipses, parabolas, and hyperbolas. Each type of conic section has distinctive properties and shapes based on the angle and position of the intersecting plane.

Hyperbolas are one of the four conic sections and are characterized by their open, curved branches that mirror each other on either side of their vertices.They differ from other conics due to their two distinct unbounded branches. Hyperbolas also have an equation that involves subtraction, as seen in the form of the equation \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\). The double set of curves seen in hyperbolas are important in various natural and technological phenomena, like in navigation systems, where their properties make solving path and position problems advantageous.