Problem 28 - Algebra and Trigonometry

Question

Find an equation for the hyperbola that satisfies the given conditions. Foci \((0, \pm 10),\) vertices \((0, \pm 8)\)

Step-by-Step Solution

Answer

The equation of the hyperbola is \(\frac{y^2}{64} - \frac{x^2}{36} = 1\).

Step 1: Identify the Type of Hyperbola

Since the foci and vertices are both on the y-axis, we determine the hyperbola is vertical.

Step 2: Determine the Center of the Hyperbola

The center of the hyperbola is the midpoint of the vertices, which is \((0, 0)\).

Step 3: Calculate the Distance c to the Foci

The distance from the center to the foci \((0, \pm 10)\) is \(c = 10\).

Step 4: Calculate the Distance a to the Vertices

The distance from the center to the vertices \((0, \pm 8)\) is \(a = 8\).

Step 5: Find the Value of b

Use the relationship \(c^2 = a^2 + b^2\) to find \(b\). Since \(c = 10\) and \(a = 8\),\[100 = 64 + b^2\]\[b^2 = 36\]\[b = 6\].

Step 6: Write the Equation of the Hyperbola

With \(a = 8\) and \(b = 6\), we insert these values into the standard form of a vertical hyperbola:\[\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\]Substituting, we get:\[\frac{y^2}{64} - \frac{x^2}{36} = 1\]

Key Concepts

Foci

In geometry, the foci of a hyperbola are two fixed points located inside each curve of the hyperbola. These points are crucial because they help define the hyperbola's shape.
For a hyperbola that opens vertically, like the one in this exercise, the foci are positioned along the y-axis.
In this example, the foci are located at

• (0, 10) and
• (0, -10)

They essentially represent the maximum distance of stretching of each curve from the center of the hyperbola.
To determine this stretching, we use the distance from the center to the foci, denoted as "c", where in this exercise, it is 10.

Vertices

The vertices of a hyperbola are the points on the curve where the hyperbola intersects its axis of symmetry. For this vertical hyperbola, these points are along the y-axis.
The given vertices are

• (0, 8) and
• (0, -8)

These points are the closest points of each hyperbola branch to the center.
The distance from the center of the hyperbola to its vertices is denoted as "a". In our example, this distance is 8.
This means the hyperbola stretches 8 units away from the center along the y-axis.

Standard Form of Hyperbola

The standard form of a hyperbola depends on its orientation, either vertically or horizontally.
For a vertical hyperbola like the one we have, the standard form is:\[\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\]In this formula:

• "a" is the distance to each vertex from the center, and
• "b" is related to the distance into the rectangle's co-vertex (this value can be easier to understand through the relationship with the foci and vertices via the equation \(c^2 = a^2 + b^2\)).

Substituting our values, we have \(a = 8\) and \(b = 6\), resulting in:\[\frac{y^2}{64} - \frac{x^2}{36} = 1\] This equation defines the boundary of our hyperbola.

Center of Hyperbola

The center of a hyperbola is the point that is equidistant from each vertex and focus.
For this exercise, the center was determined by finding the midpoint of the vertices of the hyperbola, which are located at

• (0, 8) and
• (0, -8)

Calculating the midpoint, it is quite evidently

• (0, 0)

This means the center of our hyperbola is at the origin of the coordinate plane, where it has perfect symmetry around this central point.
Locating the center is crucial for understanding the hyperbola's overall position and assuring calculations for foci and vertices are accurate.