Problem 29 - Calculus Early Transcendentals: Pearson New International Edition

Question

Hyperbola whose asymptotes are \(x \pm 2 y=0\) and that goes through the point \((4,3)\)

Step-by-Step Solution

Answer

The hyperbola's equation is \(\frac{x^2}{40} - \frac{y^2}{10} = 1\).

Step 1: Identify the Asymptotes Equations

The given asymptotes equations are \(x + 2y = 0\) and \(x - 2y = 0\). These match the asymptotes of a hyperbola centered at the origin with a horizontal transverse axis of the form\( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\).

Step 2: Determine the Relationship between a and b

For the hyperbola centered at the origin with given asymptotes, the slopes \(\pm 2\) correspond to \(\pm \frac{b}{a}\). Thus, the slopes give us \( a = 2b\).

Step 3: Use the Point to Solve for b

Substitute the point \((4, 3)\) into the equation \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\). Using \( a = 2b\), substitute to find \(b\).\[\frac{4^2}{(2b)^2} - \frac{3^2}{b^2} = 1\].

Step 4: Simplify and Solve the Equation for b

Solve the equation \(\frac{16}{4b^2} - \frac{9}{b^2} = 1\). This simplifies to \( \frac{16 - 36}{4b^2} = -1\), so \(b^2 = 10\). Therefore, \(b = \sqrt{10}\).

Step 5: Solve for a

Knowing \(a = 2b\) and \(b = \sqrt{10}\), determine \(a = 2\sqrt{10}\).

Step 6: Write the Equation of the Hyperbola

Substitute \(a\) and \(b\) back into the equation \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\). Thus, the hyperbola's equation is \(\frac{x^2}{40} - \frac{y^2}{10} = 1\).

Key Concepts

Hyperbola

A hyperbola is a fascinating type of conic section that resembles two mirror-image arcs facing away from each other. Unlike a circle or ellipse, a hyperbola is defined by a set of points where the difference in distances to two fixed points (known as foci) is constant. This unique property makes it an intriguing subject of study in calculus. Hyperbolas appear naturally in various disciplines, from physics to engineering. The standard form of a hyperbola with a center at the origin and aligned along the coordinate axes is:
\[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\]

• If the x-term is positive, the hyperbola opens left-and-right.
• If the y-term is positive, the hyperbola opens up-and-down.

Along with asymptotes, these equations define the specific shape and orientation of a hyperbola.

Asymptotes

In the context of a hyperbola, asymptotes are straight lines that the hyperbola approaches but never intersects. These lines are crucial in understanding the shape and orientation of the hyperbola. For a hyperbola centered at the origin with a horizontal transverse axis, the asymptotes are given by the equations:
\[y = \pm \frac{b}{a} x\]Asymptotes serve as a visual guide, providing the boundaries within which the curves of a hyperbola lie. They indicate where the branches of the hyperbola spread as they move outwards, giving students a clearer picture of the hyperbola's infinite extent. Asymptotes also help in sketching the graph of the hyperbola, as they define the angles at which the hyperbola's branches extend from the center. In the exercise provided, the asymptotes were identified as \(x \pm 2y = 0\), which has slopes of \(\pm 2\), essential information for determining the hyperbola's equation.

Equation of a hyperbola

The equation of a hyperbola provides a mathematical representation of its geometric structure. In its standard form, it reveals the orientation, size, and shape of the hyperbola's arcs. For the specific problem from the textbook, the equation was derived as:
\[\frac{x^2}{40} - \frac{y^2}{10} = 1\]Here is a guide to how it was formed:

• The transverse axis is aligned horizontally, so the x-term is positive.
• The values of \(a\) and \(b\) were found from the asymptotes and the point it passes through.
• The relationship \(a = 2b\) was derived from the slopes of the asymptotes.
• Substituting \(a\) and \(b\) back into the equation formed the final expression.

Understanding the components of the equation helps in visualizing how each variable affects the curve's orientation and spread.

Conic sections

Conic sections are curves obtained by intersecting a plane with a double-napped cone. They include circles, ellipses, parabolas, and hyperbolas—all integral parts of calculus research. Each conic section has its unique equation and properties that define its shape and appearance:

• A circle is formed when the plane cuts the cone parallel to its base, producing a closed curve with all points equidistant from the center.
• An ellipse, similar to a circle, has two differing axes, creating its flattened appearance.
• A parabola forms when the plane is parallel to the cone's side, creating an open curve with a clear focal point and directrix.
• A hyperbola, formed through more complex intersections, consists of two adjacent open curves.

These sections are foundational to understanding complex graph behaviors. In the provided exercise, understanding the nature of hyperbolas as a type of conic section helped solve for equations and anticipate graph behavior, highlighting calculus' role in illustrating geometric concepts.