Problem 27 - College Algebra

Question

For the following exercises, find the equations of the asymptotes for each hyperbola. $$ \frac{(x-3)^{2}}{5^{2}}-\frac{(y+4)^{2}}{2^{2}}=1 $$

Step-by-Step Solution

Answer

The asymptotes are \( y + 4 = \frac{2}{5}(x - 3) \) and \( y + 4 = -\frac{2}{5}(x - 3) \).

Step 1: Identify the hyperbola form

The given equation is \( \frac{(x-3)^{2}}{5^{2}} - \frac{(y+4)^{2}}{2^{2}} = 1 \). This corresponds to the standard form \( \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 \), which is the equation of a hyperbola with horizontal transverse axis.

Step 2: Read the values for 'a' and 'b'

From the equation, we can identify that \( a^2 = 5^2 \) and \( b^2 = 2^2 \). Thus, \( a = 5 \) and \( b = 2 \).

Step 3: Identify the center of the hyperbola

The center \((h, k)\) of the hyperbola is \((3, -4)\), extracted directly from the equation supplied as \( (x-3)^2 \) and \( (y+4)^2 \).

Step 4: Write the equations for the asymptotes

For hyperbolas of the form \( \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 \), the asymptotes are given by the equations \( y - k = \pm \frac{b}{a}(x - h) \). Substitute \( a = 5 \), \( b = 2 \), \( h = 3 \), and \( k = -4 \) into this formula to find the asymptotes.

Step 5: Calculate the asymptotes

Using the formula, the asymptote equations are: \( y + 4 = \pm \frac{2}{5}(x - 3) \). These can be rewritten as two separate linear equations: 1. \( y + 4 = \frac{2}{5}(x - 3) \)2. \( y + 4 = -\frac{2}{5}(x - 3) \).

Key Concepts

Asymptotes

Asymptotes are crucial elements in understanding the behavior of hyperbolas. In a hyperbola, asymptotes are lines that the curve approaches, but never actually reaches. They give a rough idea of how the hyperbola stretches out at infinity.

For a hyperbola, the equations of the asymptotes are derived based on the orientation of the transverse axis and the values of \( a \) and \( b \). In the case of a hyperbola with a horizontal transverse axis, the asymptotes are formulated using the expression \( y - k = \pm \frac{b}{a}(x - h) \). In this example, with \( a = 5 \), \( b = 2 \), \( h = 3 \), and \( k = -4 \), the asymptote equations become:

• \( y + 4 = \frac{2}{5}(x - 3) \)
• \( y + 4 = -\frac{2}{5}(x - 3) \)

These lines intersect at the center of the hyperbola and extend infinitely in two opposite directions, delineating the direction in which each branch of the hyperbola approaches but never intersects them.

Understanding asymptotes helps in sketching hyperbolas and predicting how they stretch out across the coordinate plane.

Horizontal Transverse Axis

The orientation of a hyperbola is determined by its transverse axis. A horizontal transverse axis indicates that the branches of the hyperbola extend left and right along the x-axis. This is evident in the given equation:

• \( \frac{(x-3)^{2}}{5^{2}} - \frac{(y+4)^{2}}{2^{2}} = 1 \)

In this standard form, the squared term with \( x \) comes before the term with \( y \), indicating that the hyperbola opens horizontally.

Here, \( a \) is associated with the x-direction, which when larger than \( b \), determines the broader stretch of the hyperbola along the transverse axis. Understanding this concept helps visualize the hyperbola and interpret its physical representation within a coordinate plane. It is also crucial when determining the equations of the asymptotes.

Center of Hyperbola

The center of a hyperbola is a crucial reference point from which its structure expands. In the standard form \( \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 \), the center is given by the point \((h, k)\). For the example equation provided, the terms \((x-3)^2\) and \((y+4)^2\) indicate:

• \( h = 3 \)
• \( k = -4 \)

Thus, the center is located at \((3, -4)\).

The center acts as a midpoint that neither branch of the hyperbola will intersect, but both will revolve around. It is also the point where the asymptotes intersect, facilitating the plotting of these lines. Knowing the center simplifies sketching a hyperbola and is instrumental in finding both its direction and scale.

Standard Form of Hyperbola

The standard form of a hyperbola helps determine its orientation, center, and asymptotes. This form is expressed as

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

For a hyperbola with a horizontal transverse axis, "\( x \)" takes precedence over "\( y \)," as seen in our example equation. In this form:

• \( a^2 \) is the denominator of the x-term and represents the axis of stretch along the transverse, impacting the horizontal direction in our case,
• \( b^2 \) is associated with the y-term, indicating the conjugate axis.

The values of \( a \) and \( b \) directly affect the shape and orientation of the hyperbola's branches. Implementing the standard form not only helps simplify calculations for the asymptotes but also sets an easy-to-follow structure for forming and manipulating hyperbola equations. Being familiar with this format can ease the process of graphing hyperbolas, leading to a clearer understanding of their characteristics and properties.