Problem 41 - Algebra 2

Question

Find the vertices and the asymptotes of each hyperbola. $$ x^{2}-9 y^{2}=1 $$

Step-by-Step Solution

Answer

The vertices of the hyperbola defined by \(x^{2} - 9y^{2} = 1\) are (1, 0) and (-1, 0). The asymptotes of the hyperbola are \(y = 3x\) and \(y = -3x\).

Step 1: Rewrite the equation to the standard form of a hyperbola.

A hyperbola equation is normally given in the form \(\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1\) for hyperbolas opening left-right or \(\frac{y^{2}}{a^{2}} - \frac{x^{2}}{b^{2}} = 1\) for those opening up-down. The given equation can be written as \(\frac{x^{2}}{1^{2}} - \frac{y^{2}}{3^{2}} = 1\), which is already in the standard form of a hyperbola. Here, \(a = 1\) and \(b = 3\).

Step 2: Find the vertices of the hyperbola.

For hyperbolas that open left-right(like our case), the vertices are at \((\pm a, 0)\). Here, a = 1, so the vertices are at \((\pm 1, 0)\). Therefore, the vertices of the hyperbola are (1, 0) and (-1, 0).

Step 3: Find the asymptotes of the hyperbola.

The equations for the asymptotes of the hyperbola are given by \(y = \pm \frac{b}{a}x\). Here, a = 1 and b = 3, so the equations for the asymptotes are \(y = \pm 3x\). Therefore, the asymptotes of the hyperbola are \(y = 3x\) and \(y = -3x\).

Key Concepts

Vertices of Hyperbolas

In a hyperbola, the vertices are the key points that lie on its principal axis and represent the closest approach of each branch to the hyperbola's center. To determine these, we first need to identify the orientation of the hyperbola, which can either open left-right or up-down.
For hyperbolas of the form \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), the hyperbola opens left-right. The vertices, in this case, are positioned at \((\pm a, 0)\).

• In the given equation \(x^2 - 9y^2 = 1\), rewritten as \(\frac{x^2}{1^2} - \frac{y^2}{3^2} = 1\), it is evident that it opens left-right.
• The value of \(a\) is \(1\), thus indicating the vertices at \((1, 0)\) and \((-1, 0)\).

These vertices serve as a reference for drawing the hyperbola, marking where it starts to curve away from the center.

Asymptotes of Hyperbolas

While the vertices mark key points on the hyperbola, the asymptotes are straight lines that aid in outlining the hyperbola's shape. Asymptotes pass through the center of the hyperbola and draw the angle at which the branches extend infinitely.
For hyperbolas with the equation \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), the equations of asymptotes are given by:
\[ y = \pm \frac{b}{a}x \]

• The equation \(x^2 - 9y^2 = 1\) reveals \(a = 1\) and \(b = 3\).
• Substituting into the asymptotes' equation, we derive \(y = \pm 3x\).

These lines aren't part of the hyperbola but guide the direction in which it opens, ensuring a symmetrical and predictable path of each branch.

Standard Form of a Hyperbola

Understanding the standard form of a hyperbola is crucial for identifying parameters such as vertices and asymptotes. The standard form provides a blueprint for determining the size and shape of a hyperbola.
Hyperbolas come in two main forms:

• Opens left-right: \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\)
• Opens up-down: \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\)

For the equation \(x^2 - 9y^2 = 1\), rewriting it as \(\frac{x^2}{1^2} - \frac{y^2}{3^2} = 1\) shows it fits the first form.

• This form makes it easier to deduce properties, like identifying that \(a = 1\) and \(b = 3\).

Being aware of these forms helps in effortlessly transitioning from raw equations to comprehensible geometric interpretations.