Problem 26
Question
Find the equations of the asymptotes for each hyperbola. \(\frac{y^{2}}{3^{2}}-\frac{x^{2}}{3^{2}}=1\)
Step-by-Step Solution
Verified Answer
The asymptotes are \( y = x \) and \( y = -x \).
1Step 1: Identify Standard Form of Hyperbola Equation
Observe the given equation of the hyperbola: \( \frac{y^2}{3^2} - \frac{x^2}{3^2} = 1 \). This matches the standard form of a vertical hyperbola \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \). Here, \( a^2 = 3^2 \) and \( b^2 = 3^2 \), so \( a = 3 \) and \( b = 3 \).
2Step 2: Write Equation of Asymptotes for Vertical Hyperbolas
For vertical hyperbolas of the form \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \), the equations of the asymptotes are given by \( y = \pm \frac{a}{b}x \).
3Step 3: Substitute Values into Asymptote Equations
Substitute \( a = 3 \) and \( b = 3 \) into the asymptote equations. This gives \( y = \pm \frac{3}{3}x \), which simplifies to \( y = \pm x \). Thus, the equations of the asymptotes are \( y = x \) and \( y = -x \).
Key Concepts
Understanding Asymptotes in a HyperbolaThe Standard Form of HyperbolasIdentifying a Vertical HyperbolaEquation of Asymptotes for Vertical Hyperbolas
Understanding Asymptotes in a Hyperbola
Asymptotes are integral to understanding hyperbolas. These are lines that the hyperbola approaches but never actually meets. Asymptotes serve as a guiding framework, indicating the potential direction and shape of the branches of the hyperbola.
For a hyperbola centered at the origin, asymptotes often cross at this point. They are important because they help visualize the extent and orientation of a hyperbola without requiring a plot.
In mathematical terms, they represent the lines that minimize the distance between themselves and the curve as the hyperbola extends infinitely.
For a hyperbola centered at the origin, asymptotes often cross at this point. They are important because they help visualize the extent and orientation of a hyperbola without requiring a plot.
In mathematical terms, they represent the lines that minimize the distance between themselves and the curve as the hyperbola extends infinitely.
The Standard Form of Hyperbolas
The standard form of a hyperbola helps in identifying whether it is vertical or horizontal. For a vertical hyperbola, the equation is given by:
\[\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\]
This form clearly shows the orientation of the hyperbola with respect to the coordinate axes. The first denominator, \( a^2 \), relates to the vertical axis (y-axis), determining the distance from the center of the hyperbola to each vertex along this axis.
By comparing an equation to this standard form, you can quickly identify key parameters such as \( a \) and \( b \), which are crucial in further analyzing the equation and graph of the hyperbola.
\[\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\]
This form clearly shows the orientation of the hyperbola with respect to the coordinate axes. The first denominator, \( a^2 \), relates to the vertical axis (y-axis), determining the distance from the center of the hyperbola to each vertex along this axis.
By comparing an equation to this standard form, you can quickly identify key parameters such as \( a \) and \( b \), which are crucial in further analyzing the equation and graph of the hyperbola.
Identifying a Vertical Hyperbola
A vertical hyperbola has its branches oriented up and down, away from the center point. The equation \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \) confirms the hyperbola opens vertically.
This specific orientation means that the major axis of the hyperbola lies on the y-axis. As such, when graphed, the branches of the hyperbola extend indefinitely along the y-axis.
Using \( y^2 \) as the leading term highlights the vertical nature since it relates to the y-axis, marking the hyperbola's orientation conclusively as vertical in this standard form.
This specific orientation means that the major axis of the hyperbola lies on the y-axis. As such, when graphed, the branches of the hyperbola extend indefinitely along the y-axis.
Using \( y^2 \) as the leading term highlights the vertical nature since it relates to the y-axis, marking the hyperbola's orientation conclusively as vertical in this standard form.
Equation of Asymptotes for Vertical Hyperbolas
The asymptotes provide clarity on the direction of a hyperbola. For vertical hyperbolas in the form of \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \), the asymptotes are straightforward lines defined by:
\[y = \pm \frac{a}{b}x \]
These equations indicate that the asymptotes pass through the origin, projecting lines that guide the hyperbola's branches. By substituting the values of \( a \) and \( b \), one can easily calculate precise asymptote equations.
For example, substituting \( a = 3 \) and \( b = 3 \) results in \( y = \pm x \), highlighting the orientation and direction of the asymptotes.
\[y = \pm \frac{a}{b}x \]
These equations indicate that the asymptotes pass through the origin, projecting lines that guide the hyperbola's branches. By substituting the values of \( a \) and \( b \), one can easily calculate precise asymptote equations.
For example, substituting \( a = 3 \) and \( b = 3 \) results in \( y = \pm x \), highlighting the orientation and direction of the asymptotes.
Other exercises in this chapter
Problem 26
For the following exercises, write the equation of an ellipse in standard form, and identify the end points of the major and minor axes as well as the foci. $$
View solution Problem 26
For the following exercises, find the equations of the asymptotes for each hyperbola. $$ \frac{y^{2}}{3^{2}}-\frac{x^{2}}{3^{2}}=1 $$
View solution Problem 27
For the following exercises, identify the conic with a focus at the origin, and then give the directrix and eccentricity. $$ r(2-\cos \theta)=1 $$
View solution Problem 27
For the following exercises, convert the polar equation of a conic section to a rectangular equation. $$ r(2-\cos \theta)=1 $$
View solution