Problem 44 - Precalculus Mathematics for Calculus

Question

The hyperbolas $$\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 \quad \text { and } \quad \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=-1$$ are said to be conjugate to each other. (a) Show that the hyperbolas \(x^{2}-4 y^{2}+16=0 \quad\) and \(\quad 4 y^{2}-x^{2}+16=0\) are conjugate to each other, and sketch their graphs on the same coordinate axes. (b) What do the hyperbolas of part (a) have in common? (c) Show that any pair of conjugate hyperbolas have the relationship you discovered in part (b).

Step-by-Step Solution

Answer

Conjugate hyperbolas share the same asymptotes.

Step 1: Rewrite the Hyperbola Equations

The given hyperbolas are \(x^2 - 4y^2 + 16 = 0\) and \(4y^2 - x^2 + 16 = 0\). Start by rearranging these equations to identify their standard form.Rearranging the first hyperbola: \(x^2 - 4y^2 = -16\) becomes \(\frac{x^2}{16} - \frac{y^2}{4} = 1\).Rearranging the second hyperbola: \(4y^2 - x^2 = -16\) becomes \(\frac{y^2}{4} - \frac{x^2}{16} = 1\).Now these equations reflect the standard forms of hyperbolas.

Step 2: Identify Conjugate Hyperbolas

Compare the rearranged equations:The first equation \(\frac{x^2}{16} - \frac{y^2}{4} = 1\) is a hyperbola opening along the x-axis.The second equation \(\frac{y^2}{4} - \frac{x^2}{16} = 1\) is a hyperbola opening along the y-axis.These equations reflect conjugate hyperbolas because their forms are "transposed": one is derived from \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) and the other from \(\frac{y^2}{b^2} - \frac{x^2}{a^2} = 1\).

Step 3: Determine Common Features

Both hyperbolas have the same asymptotes. The asymptotes for both equations \(\frac{x^2}{16} - \frac{y^2}{4} = 1\) and \(\frac{y^2}{4} - \frac{x^2}{16} = 1\) are given by the equations \( y = \pm \frac{x}{2}\). These lines serve as the boundaries within which the hyperbolas approach but never touch. This commonality defines the relationship for any pair of conjugate hyperbolas.

Step 4: Generalize to Any Pair of Conjugate Hyperbolas

For any pair of conjugate hyperbolas defined by \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) and \(\frac{y^2}{b^2} - \frac{x^2}{a^2} = 1\), the common feature is their asymptotes. These asymptotes are given by \( y = \pm \frac{b}{a}x\) and are shared between the conjugate pair. Thus, any conjugate hyperbolas will have the same asymptotes.

Key Concepts

Hyperbola Equations

Hyperbola equations are an essential part of analytic geometry. These equations describe hyperbolas, a type of conic section, which look like two mirrored curves facing away from each other. A hyperbola's equation in its standard form is represented by two squared terms, one positive and one negative, creating its unique shape. For example:

• The equation \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) represents a hyperbola that opens horizontally.
• Conversely, the equation \( \frac{y^2}{b^2} - \frac{x^2}{a^2} = 1 \) describes a hyperbola that opens vertically.

These equations render two separate curves that never actually meet, even though they seem as if they might. By rewriting any given hyperbola into this standard form, you can quickly identify its orientation and other characteristics.

Hyperbola Asymptotes

Asymptotes of a hyperbola are crucial lines that the curves get increasingly close to but never touch. Every hyperbola has two asymptotes, forming an intersecting cross through its center. These asymptotes can be derived using the formula

• For a hyperbola with equation \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), the asymptotes are \( y = \pm \frac{b}{a}x \).
• For a hyperbola with \( \frac{y^2}{b^2} - \frac{x^2}{a^2} = 1 \), the same asymptotes apply.

The presence of common asymptotes between two hyperbolas indicates that they are conjugate pairs. In simpler terms, these lines act as invisible guides, directing the paths of the hyperbolic curves in mirrored paths.

Coordinate Axes

The coordinate axes, generally represented by the x-axis and y-axis, play an important role when graphing hyperbolas. These axes provide a grid system that makes it easier to visualize hyperbolas and their properties. Here is how they interact with hyperbolas:

• For hyperbolas with equations like \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), the curves will usually open along the x-axis.
• On the other hand, \( \frac{y^2}{b^2} - \frac{x^2}{a^2} = 1 \) results in a hyperbola opening along the y-axis.

The axes thus determine the orientation and direction of the hyperbolas, ensuring that each branch faces away from each other along one of these main directions.

Standard Form of Hyperbola

The standard form of a hyperbola is crucial for understanding and identifying its properties quickly. To convert any hyperbola equation to its standard form, it should resemble one of the canonical equations:

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

Where \( a^2 \) and \( b^2 \) are constants that determine the size and shape of the hyperbola. The transformation to the standard form often involves rearranging and simplifying terms in the equation. By clearly identifying "a" and "b," the orientation (either along the x or y-axis) and aspects such as vertices, foci, and asymptotes are easily deduced. This form is not just a mathematical representation; it is a gateway to understanding the geometric nature of the hyperbola.