Problem 65 - College Algebra

Question

For the following exercises, a hedge is to be constructed in the shape of a hyperbola near a fountain at the center of the yard. Find the equation of the hyperbola and sketch the graph. The hedge will follow the asymptotes \(y=\frac{3}{4} x\) and \(y=-\frac{3}{4} x,\) and its closest distance to the center fountain is 20 yards.

Step-by-Step Solution

Answer

The equation is \( \frac{x^2}{400} - \frac{y^2}{225} = 1 \).

Step 1: Understand the asymptotes

For a hyperbola centered at the origin, the equations of the asymptotes are of the form \( y = \pm \frac{b}{a}x \). In this case, the given asymptotes are \( y = \frac{3}{4}x \) and \( y = -\frac{3}{4}x \). So, we can identify that \( \frac{b}{a} = \frac{3}{4} \).

Step 2: Identify the relationship between distance and vertices

The closest distance of the hyperbola to the center is the distance from the center to the vertex. For a hyperbola, this distance is \( a \). We are given that this distance is 20 yards, so \( a = 20 \).

Step 3: Solve for b using the asymptotes

From the relationship in Step 1, we know \( \frac{b}{a} = \frac{3}{4} \). Substituting \( a = 20 \), we find \( b = \frac{3}{4} \times 20 = 15 \).

Step 4: Write the equation of the hyperbola

Now, since the hyperbola is centered at the origin with a horizontal transverse axis, its standard equation is \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \). Substituting \( a = 20 \) and \( b = 15 \) gives us the equation: \( \frac{x^2}{400} - \frac{y^2}{225} = 1 \).

Step 5: Sketch the graph

To sketch the graph, plot the asymptotes \( y = \frac{3}{4}x \) and \( y = -\frac{3}{4}x \) first. Then, draw the hyperbola opening horizontally, passing through the points (20,0) and (-20,0) with these asymptotes as guidelines.

Key Concepts

Understanding Asymptotes in Hyperbolas

Asymptotes are crucial lines that guide the shape and direction of a hyperbola. Even though a hyperbola does not actually touch these lines, the arms of the hyperbola get infinitely close to them. In our example, the asymptotes are given by the equations \( y = \frac{3}{4}x \) and \( y = -\frac{3}{4}x \). These lines tell us that the hyperbola will extend diagonally in the same direction as the asymptotes, hinting at its orientation and spread.
Asymptotes of a hyperbola centered at the origin follow the equation \( y = \pm \frac{b}{a}x \). By comparing this to our given asymptotes, we identify the ratio \( \frac{b}{a} = \frac{3}{4} \). This relationship helps us determine the value of \( b \) once we know \( a \), which is the distance from the center to one vertex.

Locating Vertices of a Hyperbola

Vertices are essential points on a hyperbola, found at the closest distance from the center along the transverse axis. In our scenario, the hyperbola is centered at the origin, and the problem states that the closest distance is 20 yards. This means each vertex is exactly 20 units away from the center, making this distance \( a = 20 \).
Identifying the vertices of a hyperbola involves using the transverse axis. Since the hyperbola opens horizontally rooted in our asymptotic direction, the vertices' coordinates are \((20, 0)\) and \((-20, 0)\). This shows the reach of the hyperbola along the horizontal orientation and is crucial for sketching.

Determining the Distance to the Center

The distance from the center to the vertices is a fundamental aspect of hyperbolas. Known as the transverse axis length \( a \), it is the measure from the center point of a hyperbola to its furthest point horizontally (or vertically, depending on orientation). Here, we've identified \( a = 20 \) because it was given as the closest distance to the center.
Besides the transverse axis length, the relationship \( \frac{b}{a} = \frac{3}{4} \) guides us in calculating \( b \), which is the distance related to the asymptotes. Solving for \( b \), we find that \( b = 15 \), completing our set of dimensions sufficient to describe the hyperbola's spread and orientation.

Sketching a Hyperbola Graph

Sketching the graph of a hyperbola involves both mathematical understanding and a practical step-by-step approach. Start by plotting the asymptotes \( y = \frac{3}{4}x \) and \( y = -\frac{3}{4}x \). These lines intersect at the origin if the hyperbola is centered there, giving a clear guide for the hyperbola's opening direction.
Once the asymptotes are plotted, draw the vertices, which are at the points \((20, 0)\) and \((-20, 0)\). The hyperbola will shape around these vertices, opening horizontally since the transverse axis is parallel to the x-axis. Use the asymptotes as a boundary guideline to ensure the hyperbola arcs approach but never touch these lines, bowling outwards infinitely. Keep in mind the equation \( \frac{x^2}{400} - \frac{y^2}{225} = 1 \) represents the full equation to plot based on these parameters.