Problem 68

Question

Find an equation for each hyperbola. Vertices $(-10,0)$ and $(10,0)$; asymptotes $y=\pm 5 x$

Step-by-Step Solution

Verified

Answer

The equation is $\frac{x^2}{100} - \frac{y^2}{2500} = 1$.

1Step 1: Identify Center and Axes

The vertices of the hyperbola are $(-10, 0)$ and $10, 0)$. This indicates the hyperbola is centered at the origin $0, 0)$ and opens horizontally, as the x-coordinates vary.

2Step 2: Determine the Distance to Vertices

The distance from the center to a vertex is the value $a$. Since the vertices are at $(-10, 0)$ and $10, 0)$, this distance is $|10 - 0| = 10$. Thus, $a = 10$.

3Step 3: Find Asymptote Slope

The given asymptotes are $y = \pm 5x$. The slope $(\pm m)$ of the asymptotes provides information to find $b$.

4Step 4: Calculate $b$ Using Asymptote Slope

For a hyperbola opening horizontally, the relationship between $a$, $b$, and the slope $\pm m$ of the asymptotes is $\frac{b}{a} = m$. Thus, $ \frac{b}{10} = 5 \,$ giving $b = 50$.

5Step 5: Write the Hyperbola Equation

A hyperbola centered at $0, 0$ that opens horizontally has the equation $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1.$ Using $a = 10$ and $b = 50$, the equation \becomes $\frac{x^2}{100} - \frac{y^2}{2500} = 1$.

Key Concepts

Vertices of HyperbolaAsymptotesEquation of HyperbolaAxes of Hyperbola

Vertices of Hyperbola

Vertices are key features of a hyperbola, representing the points where the hyperbola intersects its transverse axis. In our exercise, the vertices are located at

(-10, 0)
(10, 0)

This indicates a horizontal hyperbola centered at the origin (0, 0).

The distance between the center and a vertex along the transverse axis is denoted by $a$. For this example:

The distance from the origin to each vertex is $|10 - 0| = 10$, so we have $a = 10$.

Asymptotes

Asymptotes of a hyperbola are straight lines that the hyperbola approaches but never touches. They provide information about the slope and orientation of the hyperbola. For the given problem, the asymptotes are:

$y = \pm 5x$

This tells us about the direction the branches of the hyperbola open, which is horizontal.

The slope $m$ of these asymptotes ($ \pm 5 $) helps us determine the value of $b$ using the relationship $\frac{b}{a} = m$. Inserting $a = 10$, we find $b = 50$.

Thus, the asymptotes guide us in understanding the "spread" of the hyperbola.

Equation of Hyperbola

The equation of a hyperbola is crucial in defining its structure mathematically. For a horizontal hyperbola centered at the origin, the standard form is:

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

Given $a = 10$ and $b = 50$, we substitute these into the equation:

\[ \frac{x^2}{100} - \frac{y^2}{2500} = 1 \]

This equation allows us to describe the precise shape and location of the hyperbola on a coordinate plane.

Axes of Hyperbola

The axes of a hyperbola are lines that help define its shape. The transverse axis is the axis along which the vertices lie, and the conjugate axis is perpendicular to it. In this case: