Problem 36 - Thomas Calculus

Question

Exercises \(35-38\) give information about the foci, vertices, and asymptotes of hyperbolas centered at the origin of the \(x y\) -plane. In each case, find the hyperbola's standard-form equation from the information given. $$ \begin{array}{l}{\text { Foci: }( \pm 2,0)} \\ {\text { Asymptotes: } y=\pm \frac{1}{\sqrt{3}} x}\end{array} $$

Step-by-Step Solution

Answer

The hyperbola's equation is \(\frac{x^2}{3} - \frac{y^2}{1} = 1\).

Step 1: Identify the Foci

The given foci for the hyperbola are located at \((\pm 2, 0)\). This means that \(c = 2\), where \(c\) is the distance from the center of the hyperbola to each focus along the transverse axis.

Step 2: Understand the Orientation

Since the foci are positioned along the x-axis, this indicates that the hyperbola opens horizontally. Therefore, the equation of the hyperbola will be of the form \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \).

Step 3: Use the Asymptotes

The equations for the asymptotes of a horizontally opening hyperbola are \( y = \pm \frac{b}{a}x \). Given \( y = \pm \frac{1}{\sqrt{3}}x \), we determine that \( \frac{b}{a} = \frac{1}{\sqrt{3}} \).

Step 4: Relate Parameters a and c

From the relationship \(c^2 = a^2 + b^2\) for hyperbolas, and knowing that \(c = 2\), we have \(4 = a^2 + b^2\).

Step 5: Solve for a and b

From Step 3, we have \( b = \frac{a}{\sqrt{3}} \). Substitute \(b\) into the equation from Step 4 to find \(a\) and \(b\).Substituting: \(4 = a^2 + \left(\frac{a}{\sqrt{3}}\right)^2\)\(4 = a^2 + \frac{a^2}{3}\)\(4 = \frac{3a^2 + a^2}{3}\)\(4 = \frac{4a^2}{3}\)\(12 = 4a^2\)\(a^2 = 3\)Therefore, \(a = \sqrt{3}\).Now use \( b = \frac{a}{\sqrt{3}} = \frac{\sqrt{3}}{\sqrt{3}} = 1\), so \(b^2 = 1\).

Step 6: Final Step: Write the Equation

Plugging \(a^2 = 3\) and \(b^2 = 1\) into the standard form equation \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) yields:\[\frac{x^2}{3} - \frac{y^2}{1} = 1\].

Key Concepts

Foci of a Hyperbola

When dealing with hyperbolas, the foci are key points that define their shape and position. In the case of a hyperbola centered at the origin, the foci lie along the transverse axis. For the hyperbola in our exercise, the foci are given as \(\pm 2, 0\). This tells us that the hyperbola's transverse axis is aligned along the \(x\)-axis, making it horizontal. The distance from the center \(0,0\) to each focus is \(c = 2\).

The relationship between the foci and other parameters of a hyperbola is governed by the equation \(c^2 = a^2 + b^2\), where \(a\) and \(b\) represent the distances related to the vertices along the transverse and conjugate axes, respectively. Identifying the exact position of the foci is crucial to solving the equation of the hyperbola.

Asymptotes of a Hyperbola

Asymptotes of a hyperbola are the lines that the hyperbola approaches but never actually reaches. For hyperbolas that open horizontally, the asymptotes serve as diagonal guidelines. In our scenario, the asymptotes are given by the equations \(y = \pm \frac{1}{\sqrt{3}} x\). These equations hint at the orientation and steepness of the lines with respect to the coordinate axes.

The equation of the asymptotes for a standard horizontal hyperbola is derived from the formula \(y = \pm \frac{b}{a}x\). By this formula, the slope \(\pm \frac{b}{a}\) for our case is \(\pm \frac{1}{\sqrt{3}}\), indicating the precise ratio of \(b\) to \(a\) for this hyperbola. Understanding and using these asymptotes are critical for defining the hyperbola's structure.

Standard Form of a Hyperbola

Every hyperbola is best described using its standard form equation, which makes the analysis of its properties much simpler. The standard form of the equation for a horizontal hyperbola is \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \). This form suits hyperbolas centered at the origin and gives immediate information on the hyperbola's stretches along the \(x\) and \(y\) axes.

In our exercise example, given that \(a^2 = 3\) and \(b^2 = 1\), the equation becomes \( \frac{x^2}{3} - \frac{y^2}{1} = 1 \). This equation encompasses all the specifics, including the size and orientation of the hyperbola. Moreover, it highlights the core geometric nature by explicitly showing the contributions of \(a\) and \(b\) to the hyperbola's shape.

Horizontal Hyperbola

A horizontal hyperbola has its transverse axis aligned along the \(x\)-axis, which influences both its general appearance and its equation. In this orientation, the hyperbola opens to the left and right sides of the center, as defined by the foci's locations. The transverse axis in a horizontal hyperbola goes along \(x\), displaying a distinct symmetry about this axis.

The equation of a horizontal hyperbola takes the form \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), with \(a\) associated with the distance from the center to the vertices along the \(x\)-axis. This is juxtaposed against a vertical hyperbola where the \(y\) terms are positive. Clearly identifying the horizontal nature of a hyperbola from the position of the foci and the given vertices' direction is imperative for forming and interpreting its equation.