Problem 40 - Algebra and Trigonometry

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 $$x^{2}-4 y^{2}+16=0 \quad \text { and } \quad 4 y^{2}-x^{2}+16=0$$ 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 have common asymptotes.

Step 1: Identify and Rewrite the Equations

First, we need to rewrite the given hyperbolas in the standard form. For the first equation, \( x^2 - 4y^2 + 16 = 0 \), rearrange it to the form \( \frac{x^2}{16} - \frac{y^2}{4} = 1 \). Similarly, rewrite the second equation \( 4y^2 - x^2 + 16 = 0 \) as \( \frac{y^2}{4} - \frac{x^2}{16} = 1 \).

Step 2: Comparison with Standard Hyperbola Form

Compare the equations obtained in Step 1 with the standard form of hyperbolas \( \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 \) and \( \frac{y^{2}}{b^{2}} - \frac{x^{2}}{a^{2}} = 1 \). Notice that \( a^2 = 16 \) and \( b^2 = 4 \) for both hyperbolas, confirming they are indeed conjugate hyperbolas.

Step 3: Sketch the Hyperbolas

To sketch the hyperbolas, note that the first hyperbola \( \frac{x^2}{16} - \frac{y^2}{4} = 1 \) has a transverse axis along the x-axis, and the second hyperbola \( \frac{y^2}{4} - \frac{x^2}{16} = 1 \) has a transverse axis along the y-axis. The center for both is at (0, 0) due to the absence of \(h\) and \(k\) terms.

Step 4: Common Characteristics

The two hyperbolas share the same asymptotes, which can be determined by setting the expressions inside the hyperbola equations to zero. For the equation \( \frac{x^2}{16} - \frac{y^2}{4} = 1 \), the asymptotes are \( y = \pm \frac{1}{2}x \). Both hyperbolas have these asymptotes, highlighting a common feature.

Step 5: General Relationship for Conjugate Hyperbolas

Observe that any pair of conjugate hyperbolas \( \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 \) and \( \frac{y^{2}}{b^{2}} - \frac{x^{2}}{a^{2}} = 1 \) will have asymptotes given by \( y = \pm \frac{b}{a}x \) and \( x = \pm \frac{a}{b}y \), which are equivalent, showing they have common asymptotes.

Key Concepts

Standard Form of Hyperbolas

The standard form of a hyperbola is an algebraic expression that represents its general equation. For hyperbolas centered at the origin, the standard form can be written as either

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

These forms are used depending on the orientation along the axes. The constants \( a^{2} \) and \( b^{2} \) are derived from the geometry of the hyperbola. In these formulas, \( a \) determines the length of the transverse axis while \( b \) determines the length of the conjugate axis.
When analyzing specific cases, like in the exercise example, rewriting an equation into one of these forms helps identify important properties, such as its center, vertices, and asymptotes.
Every time you're dealing with hyperbolas, rewriting into the standard form allows you to predict and sketch their shape on the graph easily, which is crucial in understanding their properties deeply.

Asymptotes of Hyperbolas

Asymptotes are crucial in understanding the shape and orientation of hyperbolas. These are imaginary lines that the curves of a hyperbola approach but never touch as they extend to infinity.
For hyperbolas defined by \( \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 \), the equations of the asymptotes are

• \( y = \pm \frac{b}{a}x \)

These lines cross at the center of the hyperbola, which is commonly at the origin (0,0) for standard-form hyperbolas. In the exercise, these factors help determine the commonality between the conjugate hyperbolas, as specific hyperbolas share identical asymptotes.
Recognizing asymptotes is essential to sketch the curves accurately and appreciate their geometric nature. Remember, asymptotes guide the direction in which the hyperbola's branches extend. This understanding helps in precise graphing, allowing students to portray hyperbolas accurately and appreciate the elegance in their structure.

Transverse Axis

The transverse axis is a key feature in the structure of hyperbolas. It’s the line segment that passes through the foci of the hyperbola, spanning the width between its vertices. For the hyperbolas of the form

• \( \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 \), the transverse axis is horizontal,
• \( \frac{y^{2}}{b^{2}} - \frac{x^{2}}{a^{2}} = 1 \), the transverse axis is vertical.

The length of the transverse axis is determined by \( 2a \), where \( a \) is derived from the equation's standard form. In the exercise, identifying the transverse axes allowed them to recognize which direction each hyperbola opens.
This axis is pivotal since it impacts where the graph intersects with the axes, indicating the hyperbola's reach in either the vertical or horizontal direction. Identifying the transverse axis helps in predicting the hyperbola's overall orientation and thereby acquiring a more detailed understanding of its configuration.

Conjugate Axis

Complementary to the transverse axis, the conjugate axis is also critical in detailing a hyperbola’s geometric outline. The conjugate axis is the line segment perpendicular to the transverse axis, located at the center of the hyperbola.
For hyperbolas defined by

• \( \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 \), the conjugate axis is vertical,
• \( \frac{y^{2}}{b^{2}} - \frac{x^{2}}{a^{2}} = 1 \), the conjugate axis is horizontal.

This axis typically doesn’t intersect the curve of the hyperbola but determines the steepness and how wide the hyperbola opens. Its length is quantified by \( 2b \).
Understanding the conjugate axis provides clarity on the spatial arrangement of a hyperbola, complementing the transverse axis to furnish a full picture of the hyperbola's dimensions. In conjugate hyperbolas, as in the exercise, this knowledge highlights their symmetric yet perpendicular relationship.