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 standard-form equation is \(\frac{x^2}{3} - y^2 = 1\).

Step 1: Understand the Hyperbola Structure

The foci of the hyperbola are given as \((\pm 2, 0)\), suggesting that the hyperbola opens horizontally. This reminds us that the standard form of a horizontally opening hyperbola is \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\).

Step 2: Identify the Distance to Foci

In hyperbolas, the distance from the center to each focus \(c\) is given as 2 from \((\pm 2, 0)\). Hence, \(c = 2\).

Step 3: Find the Asymptotes Relationship

The asymptotes formula for a horizontal hyperbola is \(y = \pm \frac{b}{a} x\). Given \(y = \pm \frac{1}{\sqrt{3}} x\), we see that \(\frac{b}{a} = \frac{1}{\sqrt{3}}\).

Step 4: Relate Foci and Vertices Parameters

For a hyperbola, the relationship \(c^2 = a^2 + b^2\) holds. Since \(c = 2\), we have \(c^2 = 4\).

Step 5: Solve for Parameters \(a^2\) and \(b^2\)

Using \(b = \frac{a}{\sqrt{3}}\), we have \(b^2 = \frac{a^2}{3}\). Insert this into \(c^2 = a^2 + b^2\): \[ 4 = a^2 + \frac{a^2}{3} \]Combine terms: \[ 4 = \frac{3a^2 + a^2}{3} = \frac{4a^2}{3} \]Solve for \(a^2\): \[ 4 \times 3 = 4a^2 \implies a^2 = 3 \]

Step 6: Final Calculation of \(b^2\)

Since \(b^2 = \frac{a^2}{3}\), substitute \(a^2 = 3\) to find \(b^2\):\[ b^2 = \frac{3}{3} = 1 \]

Step 7: Write the Standard Form Equation

Now that we know both \(a^2\) and \(b^2\), substitute these into the hyperbola's standard form:\[ \frac{x^2}{3} - \frac{y^2}{1} = 1 \]

Key Concepts

Standard Form of Hyperbola

A hyperbola is a type of conic section defined as the difference between the distances to two fixed points called foci is constant. The standard form of a hyperbola can tell us a lot about its orientation and key features. For horizontal hyperbolas, the formula is usually given as \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \). This formula tells us that the hyperbola opens left and right along the x-axis.

In this formula:

• \(a\) is the distance from the center of the hyperbola to each vertex on the x-axis,
• \(b\) is the distance on the y-axis associated with the curve's width,
• \(c\), the distance from the center to the foci, follows \(c^2 = a^2 + b^2\).

Using this structure, we can deduce the hyperbola's shape and the lengths of its axes. It's essential to remember these relationships, especially when matching a hyperbola to its given details, such as vertices or asymptotes, because they help in specifying the right equation that represents the hyperbola.

Asymptotes of Hyperbola

In the context of hyperbolas, asymptotes play a crucial role. They provide a guideline for the indefinite stretch of hyperbola branches. For hyperbolas, asymptotes are straight lines that pass through the center and define the general direction in which the branches open.

For a horizontal hyperbola, the asymptotes are given by the equation \(y = \pm \frac{b}{a} x\). This indicates the slope of the asymptotes based on the values of \(a\) and \(b\). In this exercise, the asymptotes are \(y = \pm \frac{1}{\sqrt{3}} x\).

Using the information from the given asymptotes, we determined that \(\frac{b}{a} = \frac{1}{\sqrt{3}}\), which is critical to finding the correct values for \(b\) and subsequently, the size and orientation of the hyperbola. Asymptotes thus offer a means to relate the geometric properties of the hyperbola back to its algebraic expression in standard form.

Foci of Hyperbola

The foci are essential features of a hyperbola, marking points about which the hyperbola is symmetrically arranged. The distance between the foci and the center is a critical factor in describing the hyperbola's energy and shape. In terms of hyperbola geometry, they're akin to the anchor points of the shape.

For the given exercise, the foci are \((\pm 2, 0)\), suggesting a horizontal opening hyperbola since the points lie along the x-axis. This implies that \(c = 2\). The relationship \(c^2 = a^2 + b^2\) helps find the other hyperbola constants once the foci locations are known.

Understanding the role of foci means you grasp one of the core elements of hyperbola configuration since they, alongside vertices, define the hyperbola's basic structure and directionality.