Problem 52 - 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

They are conjugate hyperbolas with the same center and perpendicular transverse axes.

Step 1: Simplify and Rearrange the Equations

Rewrite the given equations in standard hyperbola form by moving the constant term to the opposite side of the equation. The given equations are \(x^2 - 4y^2 + 16 = 0\) and \(4y^2 - x^2 + 16 = 0\). Simplify them to:\[x^2 - 4y^2 = -16\] \[4y^2 - x^2 = -16\]

Step 2: Express in Standard Form

Convert these equations into the standard form of hyperbolas. For \(x^2 - 4y^2 = -16\): Divide each term by \(-16\) to get \[\frac{x^2}{16} - \frac{y^2}{4} = 1\]For \(4y^2 - x^2 = -16\), divide each term by \(-16\) as well:\[\frac{y^2}{4} - \frac{x^2}{16} = 1\]

Step 3: Confirm Conjugate Relationship

Compare the forms with the standard conjugate hyperbolas \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) and \(\frac{y^2}{b^2} - \frac{x^2}{a^2} = 1\). The given equations match these forms, indicating they are conjugate hyperbolas.

Step 4: Graph the Hyperbolas

To graph, notice that both hyperbolas share a center at the origin (0,0). The hyperbola \(\frac{x^2}{16} - \frac{y^2}{4} = 1\) has its transverse axis along the x-axis, and \(\frac{y^2}{4} - \frac{x^2}{16} = 1\) has its transverse axis along the y-axis. Sketch these hyperbolas on the same coordinate axes, ensuring that each is centered at the origin.

Step 5: Discover Common Features

Both hyperbolas have the same center at the origin, and identical transverse axes ratio but in perpendicular orientation. This is typical for conjugate hyperbolas.

Step 6: General Conjugate Hyperbolas

To generalize, 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 share the same center and scale along orthogonal axes. This holds true, confirming the relationship discovered in part (b).

Key Concepts

Hyperbola Equations

Hyperbola equations are crucial to understanding how these unique curves are defined and graphed. Hyperbolas differ from ellipses as they contain two disconnected curves, often referred to as branches. These branches mirror each other. The standard form of a hyperbola equation is either \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) or \( \frac{y^2}{b^2} - \frac{x^2}{a^2} = 1 \). Both forms indicate a hyperbola centered around the origin, but the orientation of the curves differs.

In the given exercise, two specific hyperbolas were examined: \( x^2 - 4y^2 + 16 = 0 \) and \( 4y^2 - x^2 + 16 = 0 \). These equations were manipulated, reminding us of the importance of algebraic manipulation in simplifying and identifying these curves. In general, a hyperbola is defined by its relationship between its transverse and conjugate axis, and it's important to recognize these tell-tale equations when analyzing conjugate hyperbolas.

Standard Form of Hyperbola

Understanding the standard form of a hyperbola is necessary to identify and differentiate between the transverse and conjugate axes. The standard forms \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) and \( \frac{y^2}{b^2} - \frac{x^2}{a^2} = 1 \) present hyperbolas in their simplest forms. Here, \(a^2\) and \(b^2\) represent constant positive numbers that help determine the shape and orientation of a hyperbola.

In the provided solution, converting the original equations to this standard form was key to recognizing they are conjugate hyperbolas. This conversion revealed their symmetry and allowed us to graph them precisely, showing how the hyperbolas open along different axes. The manipulation to standard form not only clarifies the hyperbola's structure but also allows an easier calculation of foci, vertices, and asymptotes.

Transverse Axis

The transverse axis of a hyperbola is a key feature that defines the direction along which the branches of the hyperbola open. For hyperbolas in the form \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), the transverse axis is along the x-axis. Meanwhile, for the form \( \frac{y^2}{b^2} - \frac{x^2}{a^2} = 1 \), the transverse axis is along the y-axis.

In our specific solution, we learned that one hyperbola opens along the x-axis while its conjugate opens along the y-axis. This perpendicular orientation of their transverse axes is a distinguishing characteristic of conjugate hyperbolas. These axes pass through the vertices and are crucial for sketching the hyperbola accurately and understanding its orientation in space.