Identify The Center, Vertices, Co-vertices, Foci, And Asymptotes Of The Hyperbola Given By The Equation: ( Y + 1 ) 2 36 − ( X − 2 ) 2 4 = 1 \frac{(y+1)^2}{36} - \frac{(x-2)^2}{4} = 1 36 ( Y + 1 ) 2 ​ − 4 ( X − 2 ) 2 ​ = 1 - Center: $\qquad$- Vertices: $\qquad$- Co-Vertices: $\qquad$-

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Introduction

Hyperbolas are a type of conic section that can be used to model various real-world phenomena, such as the trajectory of a projectile or the shape of a satellite dish. In mathematics, hyperbolas are defined by their equation, which can be written in the standard form (y−k)2a2−(x−h)2b2=1\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1, where (h,k)(h,k) is the center of the hyperbola, and aa and bb are the distances from the center to the vertices and co-vertices, respectively. In this article, we will explore how to identify the center, vertices, co-vertices, foci, and asymptotes of a hyperbola given by the equation (y+1)236−(x−2)24=1\frac{(y+1)^2}{36} - \frac{(x-2)^2}{4} = 1.

Understanding the Equation

The given equation is in the standard form of a hyperbola, with the center at (h,k)=(2,−1)(h,k) = (2,-1). The values of aa and bb are 66 and 22, respectively. To identify the key components of the hyperbola, we need to understand the relationship between the equation and the graph of the hyperbola.

Identifying the Center

The center of the hyperbola is the point (h,k)(h,k) that is at the center of the hyperbola. In this case, the center is at (2,−1)(2,-1). The center is the point around which the hyperbola is symmetric.

Identifying the Vertices

The vertices of the hyperbola are the points on the hyperbola that are closest to the center. In this case, the vertices are at (h±a,k)=(2±6,−1)(h \pm a,k) = (2 \pm 6,-1). The vertices are the points where the hyperbola intersects the transverse axis.

Identifying the Co-Vertices

The co-vertices of the hyperbola are the points on the hyperbola that are farthest from the center. In this case, the co-vertices are at (h,k±a)=(2,−1±6)(h,k \pm a) = (2,-1 \pm 6). The co-vertices are the points where the hyperbola intersects the conjugate axis.

Identifying the Foci

The foci of the hyperbola are the points inside the hyperbola that are equidistant from the center. In this case, the foci are at (h±c,k)=(2±36+4,−1)(h \pm c,k) = (2 \pm \sqrt{36+4},-1), where c=a2+b2=36+4=40c = \sqrt{a^2+b^2} = \sqrt{36+4} = \sqrt{40}. The foci are the points that are inside the hyperbola and are equidistant from the center.

Identifying the Asymptotes

The asymptotes of the hyperbola are the lines that the hyperbola approaches as it goes to infinity. In this case, the asymptotes are the lines y=±ab(x−h)=±62(x−2)=±3(x−2)y = \pm \frac{a}{b}(x-h) = \pm \frac{6}{2}(x-2) = \pm 3(x-2). The asymptotes are the lines that the hyperbola approaches as it goes to infinity.

Conclusion

In conclusion, identifying the center, vertices, co-vertices, foci, and asymptotes of a hyperbola is a crucial step in understanding the properties of the hyperbola. By following the steps outlined in this article, we can identify these key components of a hyperbola given by the equation (y+1)236−(x−2)24=1\frac{(y+1)^2}{36} - \frac{(x-2)^2}{4} = 1. This knowledge can be used to model various real-world phenomena and to solve problems involving hyperbolas.

Key Takeaways

  • The center of the hyperbola is the point (h,k)(h,k) that is at the center of the hyperbola.
  • The vertices of the hyperbola are the points on the hyperbola that are closest to the center.
  • The co-vertices of the hyperbola are the points on the hyperbola that are farthest from the center.
  • The foci of the hyperbola are the points inside the hyperbola that are equidistant from the center.
  • The asymptotes of the hyperbola are the lines that the hyperbola approaches as it goes to infinity.

References

Discussion

Introduction

Hyperbolas are a fundamental concept in mathematics, and understanding their properties and characteristics is essential for solving problems in various fields. In this article, we will address some of the most frequently asked questions about hyperbolas, providing clear and concise answers to help you better understand these fascinating curves.

Q: What is a hyperbola?

A: A hyperbola is a type of conic section that is defined by its equation. It is a set of points that are equidistant from two fixed points, called foci, and is characterized by its center, vertices, co-vertices, and asymptotes.

Q: What are the key components of a hyperbola?

A: The key components of a hyperbola include:

  • Center: The point at the center of the hyperbola.
  • Vertices: The points on the hyperbola that are closest to the center.
  • Co-vertices: The points on the hyperbola that are farthest from the center.
  • Foci: The points inside the hyperbola that are equidistant from the center.
  • Asymptotes: The lines that the hyperbola approaches as it goes to infinity.

Q: How do I identify the center of a hyperbola?

A: To identify the center of a hyperbola, you need to look at the equation of the hyperbola. The center is the point (h,k) that is at the center of the hyperbola.

Q: How do I identify the vertices of a hyperbola?

A: To identify the vertices of a hyperbola, you need to look at the equation of the hyperbola. The vertices are the points (h ± a,k) that are closest to the center.

Q: How do I identify the co-vertices of a hyperbola?

A: To identify the co-vertices of a hyperbola, you need to look at the equation of the hyperbola. The co-vertices are the points (h,k ± a) that are farthest from the center.

Q: How do I identify the foci of a hyperbola?

A: To identify the foci of a hyperbola, you need to look at the equation of the hyperbola. The foci are the points (h ± c,k) that are inside the hyperbola and are equidistant from the center.

Q: How do I identify the asymptotes of a hyperbola?

A: To identify the asymptotes of a hyperbola, you need to look at the equation of the hyperbola. The asymptotes are the lines y = ± (a/b) (x-h) that the hyperbola approaches as it goes to infinity.

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

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

  • Modeling the trajectory of a projectile
  • Describing the shape of a satellite dish
  • Analyzing the motion of a pendulum
  • Studying the behavior of electrical circuits

Q: How can I use hyperbolas to solve problems in various fields?

A: Hyperbolas can be used to solve problems in various fields, including physics, engineering, and mathematics. By understanding the properties and characteristics of hyperbolas, you can use them to model and analyze complex systems and phenomena.

Conclusion

In conclusion, hyperbolas are a fundamental concept in mathematics that have many real-world applications. By understanding the key components of a hyperbola and how to identify them, you can use hyperbolas to solve problems in various fields. We hope this Q&A article has provided you with a better understanding of hyperbolas and their properties.

Key Takeaways

  • A hyperbola is a type of conic section that is defined by its equation.
  • The key components of a hyperbola include the center, vertices, co-vertices, foci, and asymptotes.
  • To identify the center of a hyperbola, look at the equation of the hyperbola.
  • To identify the vertices of a hyperbola, look at the equation of the hyperbola.
  • To identify the co-vertices of a hyperbola, look at the equation of the hyperbola.
  • To identify the foci of a hyperbola, look at the equation of the hyperbola.
  • To identify the asymptotes of a hyperbola, look at the equation of the hyperbola.

References

Discussion

What are some other real-world applications of hyperbolas? How can you use hyperbolas to solve problems in various fields? Share your thoughts and ideas in the comments below!