Problem 28 - 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 \(F(0, \pm 10)\), asymptotes \(y=\pm \frac{1}{3} x\)

Step-by-Step Solution

Answer

The hyperbola equation is \( \frac{y^2}{10} - \frac{x^2}{90} = 1 \).

Step 1: Understand the Hyperbola's Standard Form

For a hyperbola centered at the origin with vertical transverse axis, the standard form is \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \). The foci are at \( (0, \pm c) \) where \( c = \sqrt{a^2 + b^2} \).

Step 2: Use the Given Foci Information

We are told that the foci are at \( F(0, \pm 10) \), so \( c = 10 \). You can use this value in our calculations as \( c = \sqrt{a^2 + b^2} \).

Step 3: Use the Asymptotes Information

The slopes of the asymptotes for a hyperbola in the form given are \( \pm \frac{a}{b} \). Since the slopes are given as \( \pm \frac{1}{3} \), we have \( \frac{a}{b} = \frac{1}{3} \), implying \( a = \frac{b}{3} \).

Step 4: Set Up the Equations from Steps 2 and 3

From Step 2, we have \( c^2 = a^2 + b^2 = 10^2 = 100 \). From Step 3, we have \( a = \frac{b}{3} \). Substitute \( a = \frac{b}{3} \) into the first equation: \[ \left(\frac{b}{3}\right)^2 + b^2 = 100. \]

Step 5: Solve the System of Equations

First, write the equation from Step 4 as \( \frac{b^2}{9} + b^2 = 100 \). Simplify and solve: \[ \frac{b^2}{9} + \frac{9b^2}{9} = 100 \] \[ \frac{10b^2}{9} = 100 \] \[ 10b^2 = 900 \] \[ b^2 = 90. \] Use \( a = \frac{b}{3} \) to find \( a^2 = \left(\frac{\sqrt{90}}{3}\right)^2 = 10. \)

Step 6: Write the Equation of the Hyperbola

Plug the values of \( a^2 \) and \( b^2 \) into the standard form: \[ \frac{y^2}{10} - \frac{x^2}{90} = 1. \]

Step 7: Conclusion

The equation of the hyperbola with center at the origin, foci at \( F(0, \pm 10) \), and asymptotes \( y = \pm \frac{1}{3}x \) is \( \frac{y^2}{10} - \frac{x^2}{90} = 1 \).

Key Concepts

Conic Sections

Conic sections are curves that are created when a plane intersects a cone. Imagine slicing a cone at different angles, and you will end up with various shapes: circles, ellipses, parabolas, and hyperbolas.
These shapes are collectively known as conic sections. Hyperbolas, which can appear as two separate curves, are particularly interesting. They occur when the plane cuts through both the top and bottom parts of the cone.

Understanding conic sections is crucial as they are foundational in geometry and appear in many real-world applications, such as in satellite orbits and architectural designs. Each conic section has unique characteristics, like symmetry and focal points, which help us delve deeper into their mathematical properties.

Asymptotes

In the context of hyperbolas, asymptotes are lines that the hyperbola approaches but never quite reaches. They help guide the shape of the hyperbola, providing a structure in which the curves fit and extend indefinitely.
For a hyperbola centered at the origin, the equations of the asymptotes can be deduced from its standard form. The formula for asymptotes is tied to the equation of the hyperbola as the slope of the lines formed by the asymptotes is \( \pm \frac{a}{b} \).

This slope dictates the steepness of the asymptotes. In our exercise, where the asymptotes are given by \( y = \pm \frac{1}{3}x \), it tells us how the hyperbola widens or contracts as it extends away from the origin.

Foci

The foci (plural for focus) are crucial points inside each of the hyperbola's branches. They serve as 'magnets' that visually and mathematically define the shape of the hyperbola.
The basic rule for any point on the hyperbola is that the difference in distances from this point to the two foci is constant. This constant nature guides the overall form of the hyperbola, giving it a definitive structure.

For instance, in our exercise, the foci are positioned at \( F(0, \pm 10) \). These coordinates indicate a vertical stretch and inform us about the orientation and distance measure inside the hyperbola, allowing us to use these values directly in our calculations to solve for \( a \) and \( b \).

Standard Form of Hyperbola

The standard form of a hyperbola is a mathematical equation that provides a foundation for graphing and understanding the shape's properties. It is written as:\[ \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \] for hyperbolas with a vertical transverse axis.
In this setup, \( a \) and \( b \) represent the distances that help form the asymptotes and vertices. They are interrelated via the equation \( c = \sqrt{a^2 + b^2} \), with \( c \) representing the distance to each focus from the center.

Knowing the standard form aids in quickly analyzing the hyperbola's features, including its orientation and extents. In our case, by knowing \( a^2 \) and \( b^2 \) along with the other conditions, we found the equation of the hyperbola as \( \frac{y^2}{10} - \frac{x^2}{90} = 1 \). This concise form showcases the internal symmetry and structural finesse of the hyperbola.