Problem 70 - A Graphical Approach to Precalculus with Limits

Question

Find an equation for each hyperbola. Vertices \((0,5)\) and \((0,-5) ;\) passing through \((3,10)\)

Step-by-Step Solution

Answer

The equation of the hyperbola is \(\frac{y^2}{25} - \frac{x^2}{3} = 1\).

Step 1: Identify the Transverse Axis and Center

The vertices are given as \((0, 5)\) and \((0, -5)\), which means they lie on the y-axis. This indicates that the transverse axis is vertical, and the center of the hyperbola is at \((0,0)\), which is the midpoint of the vertices.

Step 2: Determine the Value of a

The distance from the center to a vertex is the value of \(a\), so \(a = 5\). The vertices are \((0, a)\) and \((0, -a)\) because they are vertically aligned.

Step 3: Write the Standard Form of the Hyperbola Equation

Since the transverse axis is vertical, the equation is of the form \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\). Substitute \(a = 5\) into the equation: \(\frac{y^2}{25} - \frac{x^2}{b^2} = 1\).

Step 4: Substitute the Point to Find b²

Use the point \((3, 10)\) which lies on the hyperbola to find \(b^2\). Substitute \(x=3\) and \(y=10\) into the equation: \(\frac{10^2}{25} - \frac{3^2}{b^2} = 1\.\)

Step 5: Solve for b²

Simplify \(\frac{100}{25} - \frac{9}{b^2} = 1\) to get \(4 - \frac{9}{b^2} = 1\). Rearrange to find \(\frac{9}{b^2} = 3\), giving \(b^2 = 3\).

Step 6: Write the Final Equation

With \(a^2 = 25\) and \(b^2 = 3\), the equation of the hyperbola is \(\frac{y^2}{25} - \frac{x^2}{3} = 1\).

Key Concepts

Transverse Axis

A hyperbola has two axes, one of which is the transverse axis. The transverse axis is the line that passes through the two vertices of the hyperbola. It is the axis along which the hyperbola opens and is considered the main axis. In this problem, since the vertices are given as

• (0, 5)
• (0, -5)

These vertices lie on the y-axis, signifying a vertical orientation. Therefore, the transverse axis is vertical in this case.
This determines the orientation of the hyperbola and defines which variable (x or y) leads the equation.

Center of a Hyperbola

The center of a hyperbola is the midpoint between its two vertices. It's an important feature because it helps us understand the symmetry of the hyperbola. To find the center, we use the formula for the midpoint between two points:
Given vertices:

• (0, 5)
• (0, -5)

The center point then is o which is

• (0,0)

The center is crucial for determining the equation of the hyperbola because it decides how we set up the standard form equation.

Standard Form of a Hyperbola

The standard form of a hyperbola's equation depends on the orientation of the transverse axis. In this case, with a vertical transverse axis, the equation is:\[\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\]Where:

• \(a^2\) is the square of the distance from the center to each vertex.
• \(b^2\) is related to the distance from the center to points on the conjugate axis.

In our example, with \(a = 5\), the equation becomes \(\frac{y^2}{25} - \frac{x^2}{b^2} = 1\).To find \(b^2\), we used the point \((3, 10)\) and solved to obtain \(b^2 = 3\). This provides us with the complete equation of the hyperbola.

Vertices of a Hyperbola

The vertices of a hyperbola are the points where each primary curve reaches its closest point to the center. They are incredibly helpful since they determine the shape and opening of the hyperbola. In this problem, the vertices are

• (0, 5)
• (0, -5)

These points show that the hyperbola will open upwards and downwards along the y-axis. The distance from the center (0,0) to any vertex is the value of\(a\), which is

• 5

Thus, the vertices directly help us write the correct hyperbola equation and understand its graph.