Problem 52
Question
REVIEW The graph of \(\left(\frac{x}{4}\right)^{2}-\left(\frac{y}{5}\right)^{2}=1\) is a hyperbola. Which set of equations represents the asymptotes of the hyperbola's graph? $$ \begin{array}{l}{\mathbf{F} \quad y=\frac{4}{5} x, y=-\frac{4}{5} x} \\\ {\mathbf{G} y=\frac{1}{4} x, y=-\frac{1}{4} x} \\ {\mathbf{H} y=\frac{5}{4} x, y=-\frac{5}{4} x} \\ {\mathbf{J} \quad y=\frac{1}{5} x, y=-\frac{1}{5} x}\end{array} $$
Step-by-Step Solution
Verified Answer
The asymptotes are represented by: \(y = \frac{5}{4}x\) and \(y = -\frac{5}{4}x\) (option H).
1Step 1: Identify Standard Form of Hyperbola Equation
The given hyperbola equation is \(\left(\frac{x}{4}\right)^{2}-\left(\frac{y}{5}\right)^{2}=1\). The standard form of a hyperbola with a horizontal transverse axis is \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\). Here, \(a = 4\) and \(b = 5\).
2Step 2: Understand Asymptote Formula for Hyperbolas
For a hyperbola in the form \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), the asymptotes are given by the equations \(y = \pm \frac{b}{a}x\). This represents the slopes of the asymptotes.
3Step 3: Calculate Asymptote Slopes
Substitute \(a = 4\) and \(b = 5\) into the asymptote formula \(y = \pm \frac{b}{a}x\). Therefore, \(y = \pm \frac{5}{4}x\) are the equations of the asymptotes.
Key Concepts
AsymptotesStandard Form of HyperbolaTransverse Axis
Asymptotes
Asymptotes are the lines that a curve approaches as it heads towards infinity. In the case of a hyperbola, the asymptotes give valuable information about the hyperbola's orientation and shape. They help to sketch the graph even with minimal information about the actual hyperbola itself.
For hyperbolas, the asymptotes act like a guide. They show how the two branches of the hyperbola stretch out. A hyperbola usually has two asymptotes, and these lines cross each other at the hyperbola's center. The slope of the asymptotes for a horizontal hyperbola is determined by the formula for the asymptotes' slope, which is given as \( y = \pm \frac{b}{a}x \).
This equation tells you that the hyperbola will approach the lines with positive and negative slopes determined by \( \frac{b}{a} \). By substituting the values of \( a \) and \( b \) from the hyperbola equation, we can find the exact asymptotes for the hyperbola. For example, in the problem given, using \( a = 4 \) and \( b = 5 \), the asymptote equations become \( y = \pm \frac{5}{4}x \). These asymptotes effectively outline the direction in which the hyperbola opens.
For hyperbolas, the asymptotes act like a guide. They show how the two branches of the hyperbola stretch out. A hyperbola usually has two asymptotes, and these lines cross each other at the hyperbola's center. The slope of the asymptotes for a horizontal hyperbola is determined by the formula for the asymptotes' slope, which is given as \( y = \pm \frac{b}{a}x \).
This equation tells you that the hyperbola will approach the lines with positive and negative slopes determined by \( \frac{b}{a} \). By substituting the values of \( a \) and \( b \) from the hyperbola equation, we can find the exact asymptotes for the hyperbola. For example, in the problem given, using \( a = 4 \) and \( b = 5 \), the asymptote equations become \( y = \pm \frac{5}{4}x \). These asymptotes effectively outline the direction in which the hyperbola opens.
Standard Form of Hyperbola
The standard form of a hyperbola's equation is crucial to understanding its properties and its graph. Hyperbolas can be thought of as open curves with two distinct branches. The standard equation provides a structured way to understand these curves.
The equation for a hyperbola with a horizontal transverse axis is \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), while for a vertical transverse axis, it is \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \). The squared terms help determine the hyperbola's orientation.
In our case, the hyperbola is in the form \( \left(\frac{x}{4}\right)^{2}-\left(\frac{y}{5}\right)^{2}=1 \), which means it has a horizontal transverse axis because the \( x^2 \) term is positive. Recognizing the positive term enables us to identify that \( a = 4 \) and \( b = 5 \), important for calculating other elements like asymptotes. The standard form gives vital parameters such as the vertices' location, the orientation, and the slope of the asymptotes.
The equation for a hyperbola with a horizontal transverse axis is \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), while for a vertical transverse axis, it is \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \). The squared terms help determine the hyperbola's orientation.
In our case, the hyperbola is in the form \( \left(\frac{x}{4}\right)^{2}-\left(\frac{y}{5}\right)^{2}=1 \), which means it has a horizontal transverse axis because the \( x^2 \) term is positive. Recognizing the positive term enables us to identify that \( a = 4 \) and \( b = 5 \), important for calculating other elements like asymptotes. The standard form gives vital parameters such as the vertices' location, the orientation, and the slope of the asymptotes.
Transverse Axis
The transverse axis of a hyperbola is essentially the axis that passes through the two foci and the center. It provides insight into the direction in which the hyperbola opens.
A key point about the transverse axis is that it determines whether the hyperbola opens horizontally or vertically. For horizontal hyperbolas, the transverse axis is along the x-axis, indicated by a positive \( x^2 \) term in the standard equation \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \). When the \( y^2 \) term is positive, the transverse axis is vertical.
Understanding the transverse axis helps in locating the center and the vertices of the hyperbola. For example, with \( a = 4 \) and \( b = 5 \) in our equation, the hyperbola's opening is mainly along the x-axis, showing its transverse axis is horizontal. This influences both the plotting of the hyperbola and its interaction with the asymptotes. The transverse axis is crucial for fully grasping how to draw and interpret hyperbolas in coordinate geometry.
A key point about the transverse axis is that it determines whether the hyperbola opens horizontally or vertically. For horizontal hyperbolas, the transverse axis is along the x-axis, indicated by a positive \( x^2 \) term in the standard equation \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \). When the \( y^2 \) term is positive, the transverse axis is vertical.
Understanding the transverse axis helps in locating the center and the vertices of the hyperbola. For example, with \( a = 4 \) and \( b = 5 \) in our equation, the hyperbola's opening is mainly along the x-axis, showing its transverse axis is horizontal. This influences both the plotting of the hyperbola and its interaction with the asymptotes. The transverse axis is crucial for fully grasping how to draw and interpret hyperbolas in coordinate geometry.
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