Problem 25 - Algebra and Trigonometry with Analytic Geometry

Question

Find an equation for the hyperbola that has its center at the origin and satisfies the given conditions. Foci \(F(0, \pm 5)\) conjugate axis of length 4

Step-by-Step Solution

Answer

The equation is \( \frac{y^2}{21} - \frac{x^2}{4} = 1 \).

Step 1: Understand the Hyperbola Equation

For a hyperbola centered at the origin with a vertical transverse axis, the equation is \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\), where \(a\) is the distance from the center to a vertex on the y-axis.

Step 2: Identify the Foci Information

The coordinates \(F(0, \pm 5)\) show that the foci are on the y-axis. Thus, \(c = 5\), where \(c\) is the distance from the center to a focus, using the relationship between extreme points of a hyperbola.

Step 3: Use Conjugate Axis Information

The conjugate axis has a length of 4, so \(2b = 4\), which means \(b = 2\). This represents the distance from the center to a vertex on the x-axis.

Step 4: Relate a, b, and c Using the Hyperbola Equation

The relationship for a hyperbola is \(c^2 = a^2 + b^2\). Substitute \(b = 2\), \(c = 5\) into the equation: \(5^2 = a^2 + 2^2\).

Step 5: Solve for a^2

Calculate \(a^2\) using the equation from Step 4: \(25 = a^2 + 4\). Thus, \(a^2 = 21\).

Step 6: Write the Final Equation

Substitute \(a^2 = 21\) and \(b^2 = 4\) into the general equation of the hyperbola to get \(\frac{y^2}{21} - \frac{x^2}{4} = 1\).

Key Concepts

Center at Origin

A hyperbola with its center at the origin ((0,0)) is positioned symmetrically about both the x-axis and the y-axis. When the center of the hyperbola is at this point, the equations become simpler due to the symmetrical placement. This positioning impacts the layout of the vertices and the axes of the hyperbola, which, in turn, affects the overall equation.
For a hyperbola with a vertical transverse axis, which means its opening is along the y-axis, the standard equation is:
\[ \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\]Here, \(a^2\) and \(b^2\) are the squares of half the lengths of the transverse and conjugate axes, respectively. Another standard form, with a horizontal transverse axis, would still have the center at the origin but the terms for x and y would switch places in the equation.

Foci

The foci of a hyperbola are pivotal points that determine its shape and orientation. Each hyperbola has two foci (plural of focus), which lie on the transverse axis. In the given problem, the foci are located at \(F(0, \pm 5)\), telling us that they are placed symmetrically along the y-axis, indicating a vertical transverse axis.

The value \(c\) represents the distance from the center of the hyperbola to each focus. In this example, \(c\) equals 5, as derived from the foci coordinates. Understanding the location of the foci helps define the extent and direction of the hyperbola's opening.

• Foci are always found at \(c\) distance from the center.
• The transverse axis determines the location of these points.
• They dictate how stretched or compressed a hyperbola appears.

Conjugate Axis

The conjugate axis of a hyperbola is perpendicular to the transverse axis and provides insight into its overall dimensions. For a hyperbola centered at the origin, with the given conjugate axis length of 4, we conclude that:

• The full length of the conjugate axis is \(2b\).
• Thus, \(2b = 4\), giving \(b = 2\).

The conjugate axis doesn't affect the actual curvature of the hyperbola but indicates how wide or tall the resulting shape can be. With the transverse axis being along the y-axis, this value of \(b\) as the half-length of the conjugate axis influences the equation part where \(x^2\) is computed as \(\frac{x^2}{b^2}\). The value \(b\) reflects the distance from the center to the co-vertices, showing how the conjugate axis encapsulates the breadth of the hyperbola.

Relationship Between a, b, c

For hyperbolas, the relationship between the semi-major axis \(a\), the semi-minor axis \(b\), and the distance to the foci \(c\) is central to determining the hyperbola's dimensions. This relationship is expressed in the equation:
\[ c^2 = a^2 + b^2\]This formula is a critical feature of hyperbolas, showcasing how \(c\) is derived from both \(a\) and \(b\).
In this problem:

• \(c = 5\), determined by foci coordinates,
• \(b = 2\), based on the conjugate axis,
• To find \(a^2\), we compute: \[5^2 = a^2 + 2^2\]\ \[25 = a^2 + 4\] \ \[a^2 = 21\]

Hence, understanding this relationship not only ensures correct values but also ensures the resulting hyperbola fits the described scenario with the center at the origin, determined foci, and specified axes.