Problem 15 - Algebra 2

Question

Write an equation for the hyperbola that satisfies each set of conditions. vertices \((-4,1)\) and \((-4,9),\) foci \((-4,5 \pm \sqrt{97})\)

Step-by-Step Solution

Answer

The equation of the hyperbola is \(\frac{(y-5)^2}{16} - \frac{(x+4)^2}{81} = 1\).

Step 1: Recognize the Orientation of the Hyperbola

Since the vertices have the same x-coordinates, the hyperbola is vertical. The general equation for a vertical hyperbola is \( \frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1 \).

Step 2: Determine the Center of the Hyperbola

The center \((h, k)\) of the hyperbola is the midpoint of the vertices \((-4,1)\) and \((-4,9)\). Thus, the center is \(\left(-4, \frac{1+9}{2}\right) = (-4,5)\).

Step 3: Determine the Distance a

The distance between the vertices \((1,9)\) is the length of the transverse axis, which is \(2a = 9 - 1 = 8\). Thus, \(a = 4\).

Step 4: Determine the Value of c

The foci are at \((-4, 5 \pm \sqrt{97})\). To find \(c\), use the distance from the center to a focus, \(c = \sqrt{97}\).

Step 5: Calculate the Value of b

Use the relationship \(c^2 = a^2 + b^2\). Substituting the known values: \((\sqrt{97})^2 = 4^2 + b^2\), simplifying yields \(97 = 16 + b^2\). Therefore, \(b^2 = 81\) and \(b = 9\).

Step 6: Write the Equation of the Hyperbola

Substitute the center \((h,k) = (-4,5)\), \(a^2 = 16\), and \(b^2 = 81\) into the equation for a vertical hyperbola:\[\frac{(y-5)^2}{16} - \frac{(x+4)^2}{81} = 1\].

Key Concepts

Vertices

In the context of a hyperbola, vertices are crucial as they define the shape and orientation of the hyperbola. They are the points where the hyperbola intersects its transverse axis. In simpler terms, the vertices are the 'outer points' of the hyperbola along the direction in which it opens.
For the given hyperbola, the vertices are

• (-4, 1)
• (-4, 9)

These vertices share the same x-coordinate, which implies that the hyperbola opens vertically. Identifying vertices helps in determining other key values like the center and the length of the transverse axis.

Vertical Hyperbola

A vertical hyperbola is one where the hyperbola opens upwards and downwards, unlike a horizontal hyperbola which opens left and right. This direction is determined by the orientation of the vertices.
For a hyperbola with vertices

• (-4, 1)
• (-4, 9)

the x-coordinates are identical, thus confirming a vertical orientation. The standard equation for a vertical hyperbola centered at

• (h, k) is
• \( \frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1 \)

where

• (h, k) is the center,
• 'a' is the distance from the center to each vertex,
and
• 'b' is related to the distance from the center to the points on the conjugate axis.

Distance Between Vertices

The distance between the vertices of a hyperbola is referred to as the length of the transverse axis. It is a crucial measure as it determines the 'width' of the hyperbola along its primary direction.
For our given hyperbola with vertices

• (-4, 1)
• (-4, 9)

the distance is simply the difference in y-coordinates, which is

• 9 - 1 = 8

This means that the value for 2a, where a is half the length of the transverse axis, is 8, giving

• a = 4.

Knowing 'a' allows us to complete the equation of the hyperbola as it is critical for determining its shape.

Foci of Hyperbola

The foci (plural of focus) of a hyperbola are points that are symmetrically located along the transverse axis. They play a role similar to the vertices but are located further out. They are important in the definition and equation of the hyperbola.
In our problem, the foci are given as

• (-4, 5 + \( \sqrt{97} \))
• (-4, 5 - \( \sqrt{97} \))

These coordinates indicate that the foci are vertically aligned with the center of the hyperbola. The distance from the center, (-4,5), to either focus is denoted as 'c', which equals \( \sqrt{97} \). This value of 'c' is essential when using the formula

• \( c^2 = a^2 + b^2 \)

to find 'b', which further refines the equation of the hyperbola.