Problem 54 - Precalculus with Calculus Previews

Question

Conjugate Hyperbolas The hyperbolas $$ \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 \quad \text { and } \quad \frac{y^{2}}{b^{2}}-\frac{x^{2}}{a^{2}}=1 $$ are said to be conjugates of each other. (a) Find an equation of the hyperbola that is conjugate to $$ \frac{x^{2}}{25}-\frac{y^{2}}{144}=1 $$ (b) Discuss how the graphs of conjugate hyperbolas are related.

Step-by-Step Solution

Answer

The conjugate hyperbola is \(\frac{y^2}{144} - \frac{x^2}{25} = 1\). Conjugate hyperbolas have axes that are perpendicular.

Step 1: Identify the Given Hyperbola

The given hyperbola is \(\frac{x^2}{25} - \frac{y^2}{144} = 1\). This can be recognized as the equation of a hyperbola in the standard form \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), where \(a^2 = 25\) and \(b^2 = 144\). Thus, \(a = 5\) and \(b = 12\).

Step 2: Write Equation of Conjugate Hyperbola

To find the equation of the conjugate hyperbola, switch the positions of the coefficients of \(x^2\) and \(y^2\) such that the hyperbola is described by \(\frac{y^2}{b^2} - \frac{x^2}{a^2} = 1\). Substitute the values previously found, \(a^2 = 25\) and \(b^2 = 144\), into this new form to get \(\frac{y^2}{144} - \frac{x^2}{25} = 1\).

Step 3: Recognize the Relationship of Graphs

The graphs of conjugate hyperbolas have their axes of symmetry oriented perpendicular to each other. For \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), the hyperbola opens along the x-axis. For \(\frac{y^2}{b^2} - \frac{x^2}{a^2} = 1\), the hyperbola opens along the y-axis.

Key Concepts

Hyperbola

A hyperbola is a type of conic section, which looks similar to two mirrored curves opening in opposite directions. It's defined as the set of all points such that the absolute difference between the distances to two fixed points, called foci, is constant. Hyperbolas can appear in various orientations and sizes depending on the values within its equation. There are some key components:

• The center, which is the midpoint between the two foci.
• The vertices, which are the closest points on each branch to the center.
• The asymptotes, which are lines that the hyperbola approaches but never actually intersects.

A hyperbola's general form is represented by the equation:\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]Here, the hyperbola opens along the x-axis. Another form of the hyperbola is:\[ \frac{y^2}{b^2} - \frac{x^2}{a^2} = 1 \] This signifies that the hyperbola opens along the y-axis. Understanding hyperbolas involves recognizing these components and their role in defining its shape.

Standard Form

The standard form of a hyperbola is crucial for understanding and graphing it. This form provides a structured way to identify the key features of the hyperbola. In particular, it helps quickly determine:

• The center of the hyperbola. In a standard form \( \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 \), \((h,k)\) represents the center.
• Values \(a\) and \(b\), which affect the distances from the center to the vertices along the x and y axes respectively.
• The orientation of the hyperbola based on whether \(x^2\) or \(y^2\) appears first. This also determines along which axis the hyperbola opens.

For example, in the equation \( \frac{x^2}{25} - \frac{y^2}{144} = 1 \), we classically recognize it as a hyperbola in the standard form. Here, \(a^2\) is 25 and \(b^2\) is 144, with the hyperbola opening along the x-axis. It's not just a calculational tool but a visualization aid in gauging the shape and orientation of the hyperbola in geometrical space.

Axes of Symmetry

The axes of symmetry are fundamental in determining how the branches of a hyperbola are oriented in relation to the Cartesian plane. A hyperbola will have two axes:

• The transverse axis, which is where the vertices of the hyperbola are located. This is the axis along which the hyperbola's branches open. For the equation \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), the transverse axis is the x-axis.
• The conjugate axis, which runs perpendicular to the transverse axis. While it isn’t a line that the hyperbola directly touches, it suggests the width of the gap between the branches perpendicular to the transverse axis. In the case of \( \frac{y^2}{b^2} - \frac{x^2}{a^2} = 1 \), the role of transverse and conjugate axes switches, so the y-axis becomes the transverse axis.

Conjugate hyperbolas illustrate these axes beautifully. With each switch in variable dominance (from \(x\) first to \(y\) first), the primary opening direction of the hyperbola changes, reflecting of how these axes influence the shape and orientation of each conjugate hyperbola. Recognizing the axes of symmetry aids in not just graphing the hyperbola correctly but also in understanding the spatial dynamics of its orientation.