Problem 29 - 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. \(x\)-intercepts \(\pm 5, \quad\) asymptotes \(y=\pm 2 x\)

Step-by-Step Solution

Answer

\(\frac{x^2}{25} - \frac{y^2}{100} = 1\).

Step 1: Understanding the Hyperbola's Center and Axes

The hyperbola's center is at the origin, (0,0). For a hyperbola centered at the origin, the general equation is \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) or \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\), depending on the orientation. Since the x-intercepts are given, it's a horizontally-oriented hyperbola, so use \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\).

Step 2: Determine the Value of 'a' Using X-intercepts

The x-intercepts are \(\pm 5\). This means when \(y=0\), \(x=\pm 5\). Substitute into the equation, \(\frac{x^2}{a^2} = 1\):\[\frac{5^2}{a^2} = 1\]Thus, \(a^2 = 25\), and so \(a = 5\).

Step 3: Using Asymptotes to Find 'b'

The asymptotes for a hyperbola of the form \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) are given by \(y = \pm \frac{b}{a}x\). Here, \(y = \pm 2x\) implies \(\frac{b}{a} = 2\). Since \(a = 5\), we get \(b = 10\), so \(b^2 = 100\).

Step 4: Writing the Equation

With \(a^2 = 25\) and \(b^2 = 100\), substitute back into the standard form \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\):\[\frac{x^2}{25} - \frac{y^2}{100} = 1\]. This is the equation of the hyperbola.

Key Concepts

Understanding the Hyperbola

A hyperbola is a fascinating type of curve found in mathematics, and it consists of two separate branches that mirror each other. At its core, a hyperbola is defined by an equation that dictates its shape and orientation. The standard form of the equation for a hyperbola centered at the origin is

• Horizontally oriented: \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\)
• Vertically oriented: \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\)

For hyperbolas, the signs in the equations are opposite such that one part is positive and the other negative. This is different from ellipses where both are positive.
In this exercise, we have a horizontally oriented hyperbola because the x-intercepts are given, which means the curve opens to the left and right.

Decoding the Asymptotes

Asymptotes are lines that a curve approaches but never actually meets. For hyperbolas, asymptotes play a crucial role in dictating the orientation and shape of the curve.
The formula for the asymptotes of a hyperbola, when centered at the origin, is derived from the orientation of the hyperbola:

• For \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), the asymptotes are \(y = \pm \frac{b}{a}x\)
• For \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\), the asymptotes are \(y = \pm \frac{a}{b}x\)

Given asymptotes \(y = \pm 2x\), we deduce that \(\frac{b}{a} = 2\). With \(a = 5\), it leads to \(b = 10\). These equations define the slope of the lines the hyperbola approaches, providing a 'guide' of its width and orientation.

Exploring the X-intercepts

X-intercepts are where the hyperbola crosses the x-axis, and they tell us important information about the hyperbola's basic structure. Given x-intercepts of \(\pm 5\), it simply means these points satisfy the hyperbola equation when the y-coordinate is zero.
To find the value of \(a\), we substitute \(x = 5\) into the equation \(\frac{x^2}{a^2} = 1\), yielding \(a^2 = 25\). Therefore, \(a = 5\). The x-intercepts allow us to learn how far the branches of the hyperbola extend horizontally, essentially providing the half-length of the hyperbola's transverse axis.

Locating the Center at the Origin

The center of a hyperbola is a key point around which the symmetry of the curve is organized. When the center of the hyperbola is at the origin, (\(0,0\)), it simplifies calculations and the standard form equations. This centering makes the hyperbola equation more straightforward to work with, as shown in our exercise.
Centering the hyperbola at the origin gives the neat template equation for the curve, in this case, \(\frac{x^2}{25} - \frac{y^2}{100} = 1\). This creates a balanced equation around the origin, making this a symmetrical setup where calculations revolve more conveniently around \(x=0\) and \(y=0\).