Problem 33
Question
Find an equation for the hyperbola that has its center at the origin and satisfies the given conditions. \(x\) -intercepts \(\pm 5, \quad\) asymptotes \(y=\pm 2 x\)
Step-by-Step Solution
Verified Answer
The equation of the hyperbola is \( \frac{x^2}{25} - \frac{y^2}{100} = 1 \).
1Step 1: Understand the General Equation
The general equation for a hyperbola centered at the origin is \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) if it opens horizontally, or \( \frac{y^2}{b^2} - \frac{x^2}{a^2} = 1 \) if it opens vertically. Here, the given asymptotes help us determine the orientation.
2Step 2: Determine the Orientation and Configurations
Given the asymptotes are \( y = \pm 2x \), it indicates a horizontally opening hyperbola. Thus, we use \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \). Here, the slopes of the asymptotes are equal to \( \frac{b}{a} \). Hence, \( \frac{b}{a} = 2 \).
3Step 3: Find the Value of \(a\)
We know the \(x\)-intercepts are \( \pm 5 \). For a hyperbola with intercepts at these points, \(a = 5\). Thus, \(a^2 = 25\).
4Step 4: Calculate \(b\) Using Asymptotes
Using \( \frac{b}{a} = 2 \) and \(a = 5\), we find \(b = 2 \times 5 = 10\). Therefore, \(b^2 = 100\).
5Step 5: Write the Equation
Now substitute the known values of \(a\) and \(b\) into the equation of the hyperbola. Thus, the equation becomes \( \frac{x^2}{25} - \frac{y^2}{100} = 1 \).
Key Concepts
Understanding ConicsExploring Asymptotes of HyperbolasFinding the X-intercepts
Understanding Conics
Conics, or conic sections, are a group of curves formed by the intersection of a plane with a double cone. These curves include circles, ellipses, parabolas, and hyperbolas. Conics are essential in mathematics because they help describe various natural and man-made phenomena.
For instance, when the plane cuts through both nappes (sides) of the cone, the resulting shape is a hyperbola, which is the type of conic we are dealing with in this exercise. Each type of conic section has distinctive characteristics defined by its equation. Hyperbolas are open curves with two branches and are described by the equation \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) if they open horizontally. This equation changes if the hyperbola opens vertically.
Conics can have a lot of applications, such as in the orbits of celestial bodies and the design of optical instruments. Understanding their properties helps in analyzing systems that can be explained using these curves.
For instance, when the plane cuts through both nappes (sides) of the cone, the resulting shape is a hyperbola, which is the type of conic we are dealing with in this exercise. Each type of conic section has distinctive characteristics defined by its equation. Hyperbolas are open curves with two branches and are described by the equation \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) if they open horizontally. This equation changes if the hyperbola opens vertically.
Conics can have a lot of applications, such as in the orbits of celestial bodies and the design of optical instruments. Understanding their properties helps in analyzing systems that can be explained using these curves.
Exploring Asymptotes of Hyperbolas
Asymptotes are lines that a curve approaches as it heads towards infinity but never actually meets. For hyperbolas, asymptotes play a crucial role as they help to define the orientation and shape of the curve. In the given problem, the hyperbola has asymptotes at \( y = \pm 2x \), indicating a horizontally opening hyperbola.
These lines guide the arms of the hyperbola and tell us about the ratio of \( b \) to \( a \) (where \( \frac{b}{a} = 2 \) in this case), which ultimately determines the angle at which the branches spread. Understanding the relationship between the hyperbola and its asymptotes is key to correctly graphing and solving these problems.
These lines guide the arms of the hyperbola and tell us about the ratio of \( b \) to \( a \) (where \( \frac{b}{a} = 2 \) in this case), which ultimately determines the angle at which the branches spread. Understanding the relationship between the hyperbola and its asymptotes is key to correctly graphing and solving these problems.
- Asymptotes give us information about the steepness or the flatness of the curves.
- The equations of the lines \( y = \pm mx \) help us establish the relationship \( m = \frac{b}{a} \), where \( m \) is the slope.
- For our hyperbola, with \( m = 2 \), it indicates a considerable opening.
Finding the X-intercepts
X-intercepts are points where a curve or function crosses the x-axis. For hyperbolas, these points are critical, as they help define the size and orientation of the curve. In this problem, the given x-intercepts are \( \pm 5 \), which informs us that the value of \( a \) should be 5, since \( a \) represents the distance from the center to each x-intercept.
Finding x-intercepts involves setting \( y = 0 \) in the hyperbola's equation and solving for \( x \). This results in \( x = \pm a \).
Finding x-intercepts involves setting \( y = 0 \) in the hyperbola's equation and solving for \( x \). This results in \( x = \pm a \).
- In the equation \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), x-intercepts appear when \( x^2 = a^2 \), yielding \( x = \pm a \).
- For this exercise, we know \( a = 5 \), making the intercepts \( x = \pm 5 \).
- X-intercepts are useful for validating the correctness of the hyperbola equation derived.
Other exercises in this chapter
Problem 32
Find an equation of the parabola that satisfies the given conditions. $$\text { Vertex } V(4,7), \quad \text { focus } F(4,2)$$
View solution Problem 33
Find a polar equation of the parabola with focus at the pole and the given vertex. $$V\left(4, \frac{\pi}{2}\right)$$
View solution Problem 33
Find a polar equation that has the same graph as the equation in \(x\) and \(y\). $$x^{2}+(y+3)^{2}=9$$
View solution Problem 33
Exer \(19-36:\) Find an equation for the ellipse that has its center at the origin and satisfies the given conditions. \(x\) -intercepts \(\pm 2\) \(y\) -interc
View solution