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

In mathematics, a hyperbola is a type of curve that is defined by a set of 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 explore how to find the equation of a hyperbola that is centered at the origin and has foci at (-4,0) and (4,0), and vertices at (-3,0) and (3,0).

Understanding the Basics of Hyperbolas

A hyperbola is a two-dimensional curve that is defined by a set of points that satisfy a specific equation. The general equation of a hyperbola is given by:

$$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$

where $a$ and $b$ are constants that determine the shape and size of the hyperbola.

Identifying the Center and Foci of the Hyperbola

The center of the hyperbola is the point that is equidistant from the two foci. In this case, the center of the hyperbola is at the origin (0,0).

The foci of the hyperbola are the two points that are fixed and are used to define the curve. In this case, the foci are at (-4,0) and (4,0).

Identifying the Vertices of the Hyperbola

The vertices of the hyperbola are the two points that are closest to the center and are used to define the curve. In this case, the vertices are at (-3,0) and (3,0).

Finding the Equation of the Hyperbola

To find the equation of the hyperbola, we need to use the information we have about the center, foci, and vertices.

First, we need to find the value of $a$, which is the distance from the center to the vertices. We can do this by using the distance formula:

$$a = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

where $(x_1, y_1)$ is the center of the hyperbola and $(x_2, y_2)$ is one of the vertices.

Plugging in the values, we get:

$$a = \sqrt{(3 - 0)^2 + (0 - 0)^2} = 3$$

Next, we need to find the value of $c$, which is the distance from the center to the foci. We can do this by using the distance formula:

$$c = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

where $(x_1, y_1)$ is the center of the hyperbola and $(x_2, y_2)$ is one of the foci.

Plugging in the values, we get:

$$c = \sqrt{(4 - 0)^2 + (0 - 0)^2} = 4$$

Now that we have the values of $a$ and $c$, we can use the equation:

$$c^2 = a^2 + b^2$$

to find the value of $b$.

Plugging in the values, we get:

$$4^2 = 3^2 + b^2$$

Simplifying, we get:

$$16 = 9 + b^2$$

Subtracting 9 from both sides, we get:

$$7 = b^2$$

Taking the square root of both sides, we get:

$$b = \sqrt{7}$$

Now that we have the values of $a$ and $b$, we can write the equation of the hyperbola:

$$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$

Plugging in the values, we get:

$$\frac{x^2}{9} - \frac{y^2}{7} = 1$$

Conclusion

In this article, we have explored how to find the equation of a hyperbola that is centered at the origin and has foci at (-4,0) and (4,0), and vertices at (-3,0) and (3,0). We have used the information about the center, foci, and vertices to find the values of $a$ and $b$, and then used these values to write the equation of the hyperbola.

References

• [1] "Hyperbola" by Math Open Reference. Retrieved February 2023.
• [2] "Hyperbola" by Khan Academy. Retrieved February 2023.

Additional Resources

• [1] "Hyperbola" by Wolfram MathWorld. Retrieved February 2023.
• [2] "Hyperbola" by Brilliant. Retrieved February 2023.
  Hyperbola Q&A: Frequently Asked Questions =============================================

Introduction

In our previous article, we explored how to find the equation of a hyperbola that is centered at the origin and has foci at (-4,0) and (4,0), and vertices at (-3,0) and (3,0). In this article, we will answer some of the most frequently asked questions about hyperbolas.

Q: What is a hyperbola?

A: A hyperbola is a type of curve that is defined by a set of points such that the absolute value of the difference between the distances from two fixed points (called foci) is constant.

Q: What are the foci of a hyperbola?

A: The foci of a hyperbola are the two points that are fixed and are used to define the curve. In a hyperbola centered at the origin, the foci are located on the x-axis.

Q: What are the vertices of a hyperbola?

A: The vertices of a hyperbola are the two points that are closest to the center and are used to define the curve. In a hyperbola centered at the origin, the vertices are located on the x-axis.

Q: How do I find the equation of a hyperbola?

A: To find the equation of a hyperbola, you need to know the coordinates of the foci and the vertices. You can use the information about the center, foci, and vertices to find the values of $a$ and $b$, and then use these values to write the equation of the hyperbola.

Q: What is the difference between a hyperbola and an ellipse?

A: A hyperbola and an ellipse are both types of curves that are defined by a set of points. However, the key difference between the two is that a hyperbola has two separate branches, while an ellipse has a single, continuous curve.

Q: Can a hyperbola be centered at any point?

A: Yes, a hyperbola can be centered at any point. However, the foci and vertices of the hyperbola will be located at a distance of $c$ and $a$ from the center, respectively.

Q: How do I graph a hyperbola?

A: To graph a hyperbola, you need to know the equation of the hyperbola. You can use a graphing calculator or a computer program to graph the hyperbola.

Q: What are some real-world applications of hyperbolas?

A: Hyperbolas have many real-world applications, including:

• Physics: Hyperbolas are used to describe the motion of objects under the influence of a central force.
• Engineering: Hyperbolas are used to design curves for bridges, roads, and other structures.
• Computer Science: Hyperbolas are used in computer graphics to create realistic images.

Conclusion

In this article, we have answered some of the most frequently asked questions about hyperbolas. We hope that this information has been helpful in understanding the concept of hyperbolas and how they are used in mathematics and real-world applications.

References

• [1] "Hyperbola" by Math Open Reference. Retrieved February 2023.
• [2] "Hyperbola" by Khan Academy. Retrieved February 2023.

Additional Resources

• [1] "Hyperbola" by Wolfram MathWorld. Retrieved February 2023.
• [2] "Hyperbola" by Brilliant. Retrieved February 2023.