Problem 70 - A Graphical Approach to College Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify the Center from Vertices

The vertices (0,5) and (0,-5) give us information about the position and size of the hyperbola. Since both vertices lie on the y-axis, we know the hyperbola is vertical. The center of the hyperbola is the midpoint of (0,5) and (0,-5), which is (0,0).

Step 2: Determine the Parameter a

The distance between the center (0,0) and either vertex (0,5) or (0,-5) is 5. In a hyperbola, this distance is represented by the parameter 'a'. Therefore, a = 5.

Step 3: Write the General Equation of the Hyperbola

For a vertical hyperbola centered at (0,0), the standard form of the equation is: \[ \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \]Substituting a = 5, we have: \[ \frac{y^2}{25} - \frac{x^2}{b^2} = 1 \]

Step 4: Use the Given Point to Solve for b

The hyperbola also passes through the point (3,10). Substitute x=3, y=10 into the equation: \[ \frac{10^2}{25} - \frac{3^2}{b^2} = 1 \]\[ \frac{100}{25} - \frac{9}{b^2} = 1 \] \[ 4 - \frac{9}{b^2} = 1 \] \[ \frac{9}{b^2} = 3 \] \[ b^2 = 3 \] Thus, b is the square root of 3.

Step 5: Write the Final Equation of the Hyperbola

Now that we have determined b^2 = 3, substitute this back into the hyperbola equation: \[ \frac{y^2}{25} - \frac{x^2}{3} = 1 \] This is the equation of the hyperbola.

Key Concepts

Vertices

Vertices are two specific points on a hyperbola, situated along its axis. They are critical in determining the shape and orientation of the hyperbola.
Understanding the location of vertices helps in sketching the structure of the hyperbola and provides insight into its dimensions.

Here, the vertices are \( (0,5) \) and \( (0,-5) \), indicating a vertical hyperbola.

• The vertices lie along the y-axis, confirming the orientation.
• From the vertices, you can determine the direction: vertical hyperbolas open upward and downward.

Since these points are equidistant from the center of the hyperbola, the midpoint can be easily calculated, aiding in locating the center.

Center of Hyperbola

The center of a hyperbola is the midpoint between its vertices. It acts as the equilibrium point from which the hyperbola stretches out in both directions. In geometric terms, the center is essentially the average point between the vertices.

In the given exercise, the vertices \( (0,5) \) and \( (0,-5) \) pin the center down at \( (0,0) \).

• The center is the starting point for both the transverse and conjugate axes.
• It serves as the origin for measuring distances such as 'a' or 'b'.

Having the center readily identified makes it easier to apply the standard form equation of the hyperbola. Knowing the center helps in understanding and defining both axes clearly.

Standard Form of Hyperbola

The standard form of a hyperbola's equation is crucial for expressing its characteristics mathematically. A vertical hyperbola centered at the origin has its equation defined as:

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

This equation's structure captures how the y-term and x-term relate to the parameters 'a' and 'b' respectively.

• 'a' relates to the distance from the center to the vertices along the vertical axis.
• 'b' describes the distance from the center along the horizontal axis, though it's not the foremost determining feature for the hyperbola's opening.

This structure is indispensable because it succinctly characterizes a hyperbola's spread based on its centered origin and its relationship with 'a' and 'b' parameters.

Parameter a

Within a hyperbola, the parameter 'a' signifies the distance from the center to each vertex along the transverse axis. It essentially controls how the hyperbola stretches perpendicular to the conjugate axis.

Using the exercise example, the given vertices \( (0,5) \) and \( (0,-5) \) let us determine that:\( a = 5 \).

• 'a' is always a positive value as it denotes span, regardless of direction.
• In the equation, it determines the denominator under the y-term for vertical hyperbolas.

Understanding 'a' is pivotal because it provides the precise measurement needed to anchor the hyperbola's equation. Calculating 'a' correctly ensures that the opening of the hyperbola aligns accurately with the features specified in the problem.