This Hyperbola Is Centered At The Origin. Find Its Equation.Foci: { (-4,0)$}$ And { (4,0)$}$ Vertices: { (-3,0)$}$ And { (3,0)$}$ ${\frac{x^2}{9} - \frac{y^2}{7} = 1}$
Introduction
A hyperbola is a type of conic section that consists of two separate branches. It is defined as the set of all points such that the absolute value of the difference between the distances from two fixed points (called foci) is constant. In this article, we will focus on finding the equation of a hyperbola that is centered at the origin and has its foci and vertices given.
Understanding the Given Information
We are given the following information about the hyperbola:
- Foci: (-4, 0) and (4, 0)
- Vertices: (-3, 0) and (3, 0)
From this information, we can determine the center of the hyperbola, which is the midpoint of the line segment connecting the foci. Since the foci are (-4, 0) and (4, 0), the center is at the origin (0, 0).
Determining the Transverse Axis
The transverse axis of a hyperbola is the line segment that passes through the center and is perpendicular to the conjugate axis. In this case, the transverse axis is the x-axis, since the foci and vertices lie on the x-axis.
Finding the Distance between the Foci
The distance between the foci is given by the formula:
2c = distance between foci
where c is the distance from the center to each focus. In this case, the distance between the foci is 8, so we have:
2c = 8 c = 4
Finding the Distance between the Vertices
The distance between the vertices is given by the formula:
2a = distance between vertices
where a is the distance from the center to each vertex. In this case, the distance between the vertices is 6, so we have:
2a = 6 a = 3
Finding the Equation of the Hyperbola
The equation of a hyperbola with a transverse axis along the x-axis is given by:
(x2/a2) - (y2/b2) = 1
where a is the distance from the center to each vertex, and b is the distance from the center to each co-vertex. We have already found the value of a, which is 3.
To find the value of b, we can use the formula:
b^2 = c^2 - a^2
where c is the distance from the center to each focus. We have already found the value of c, which is 4. Plugging in the values, we get:
b^2 = 4^2 - 3^2 b^2 = 16 - 9 b^2 = 7
So, the equation of the hyperbola is:
(x^2/9) - (y^2/7) = 1
Conclusion
In this article, we have found the equation of a hyperbola that is centered at the origin and has its foci and vertices given. We have used the given information to determine the center, transverse axis, and distances between the foci and vertices. We have then used these values to find the equation of the hyperbola.
Key Takeaways
- The equation of a hyperbola with a transverse axis along the x-axis is given by (x2/a2) - (y2/b2) = 1.
- The distance between the foci is given by 2c, where c is the distance from the center to each focus.
- The distance between the vertices is given by 2a, where a is the distance from the center to each vertex.
- The value of b can be found using the formula b^2 = c^2 - a^2.
Further Reading
For more information on hyperbolas, including their properties and equations, see the following resources:
References
- [Boyer, C. B. (1985). A History of Mathematics. New York: Wiley.]
- [Kline, M. (1972). Mathematical Thought from Ancient to Modern Times. New York: Oxford University Press.]
Hyperbola Q&A: Frequently Asked Questions =============================================
Introduction
In our previous article, we discussed how to find the equation of a hyperbola that is centered at the origin and has its foci and vertices given. In this article, we will answer some frequently asked questions about hyperbolas.
Q: What is a hyperbola?
A: A hyperbola is a type of conic section that consists of two separate branches. It is defined as the set of all points such that the absolute value of the difference between the distances from two fixed points (called foci) is constant.
Q: What are the key components of a hyperbola?
A: The key components of a hyperbola are:
- Center: The point at the center of the hyperbola.
- Transverse axis: The line segment that passes through the center and is perpendicular to the conjugate axis.
- Conjugate axis: The line segment that passes through the center and is parallel to the transverse axis.
- Foci: The two fixed points that define the hyperbola.
- Vertices: The two points on the transverse axis that are closest to the foci.
Q: How do I find the equation of a hyperbola?
A: To find the equation of a hyperbola, you need to know the following information:
- Center: The point at the center of the hyperbola.
- Transverse axis: The line segment that passes through the center and is perpendicular to the conjugate axis.
- Distance between foci: The distance between the two foci.
- Distance between vertices: The distance between the two vertices.
You can use the following formulas to find the equation of the hyperbola:
- a: The distance from the center to each vertex.
- b: The distance from the center to each co-vertex.
- c: The distance from the center to each focus.
The equation of a hyperbola with a transverse axis along the x-axis is given by:
(x2/a2) - (y2/b2) = 1
Q: What is the difference between a hyperbola and an ellipse?
A: A hyperbola and an ellipse are both types of conic sections, but they have some key differences:
- Shape: A hyperbola has two separate branches, while an ellipse is a closed curve.
- Foci: A hyperbola has two foci, while an ellipse has two foci that are on the same line.
- Equation: The equation of a hyperbola is different from the equation of an ellipse.
Q: Can I use a hyperbola to model real-world phenomena?
A: Yes, hyperbolas can be used to model real-world phenomena such as:
- Projectile motion: The trajectory of a projectile under the influence of gravity can be modeled using a hyperbola.
- Sound waves: The propagation of sound waves can be modeled using a hyperbola.
- Electromagnetic waves: The propagation of electromagnetic waves can be modeled using a hyperbola.
Q: How do I graph a hyperbola?
A: To graph a hyperbola, you need to know the following information:
- Center: The point at the center of the hyperbola.
- Transverse axis: The line segment that passes through the center and is perpendicular to the conjugate axis.
- Distance between foci: The distance between the two foci.
- Distance between vertices: The distance between the two vertices.
You can use the following steps to graph a hyperbola:
- Draw the transverse axis: Draw a line segment that passes through the center and is perpendicular to the conjugate axis.
- Draw the conjugate axis: Draw a line segment that passes through the center and is parallel to the transverse axis.
- Draw the foci: Draw the two foci on the transverse axis.
- Draw the vertices: Draw the two vertices on the transverse axis.
- Draw the branches: Draw the two branches of the hyperbola.
Conclusion
In this article, we have answered some frequently asked questions about hyperbolas. We have discussed the key components of a hyperbola, how to find the equation of a hyperbola, and how to graph a hyperbola. We have also discussed the differences between a hyperbola and an ellipse, and how hyperbolas can be used to model real-world phenomena.
Key Takeaways
- A hyperbola is a type of conic section that consists of two separate branches.
- The key components of a hyperbola are the center, transverse axis, conjugate axis, foci, and vertices.
- To find the equation of a hyperbola, you need to know the center, transverse axis, distance between foci, and distance between vertices.
- A hyperbola can be used to model real-world phenomena such as projectile motion, sound waves, and electromagnetic waves.
Further Reading
For more information on hyperbolas, including their properties and equations, see the following resources:
References
- [Boyer, C. B. (1985). A History of Mathematics. New York: Wiley.]
- [Kline, M. (1972). Mathematical Thought from Ancient to Modern Times. New York: Oxford University Press.]