Select The Correct Answer.Which Hyperbola Has Both Foci Lying In The Same Quadrant?A. $\frac{(x-24)^2}{24^2}-\frac{(y-1)^2}{7^2}=1$B. $\frac{(y-12)^2}{5^2}-\frac{(x-6)^2}{12^2}=1$C.

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Introduction to Hyperbolas

Hyperbolas are a type of mathematical curve that consists of two separate branches, resembling a pair of mirror-image curves. They are defined by a set of points that have a constant difference between the distances to two fixed points, known as the foci. In this article, we will delve into the world of hyperbolas and explore the concept of foci lying in the same quadrant.

What are Foci in Hyperbolas?

Foci are the two fixed points that define a hyperbola. They are located on the transverse axis, which is the axis that is perpendicular to the conjugate axis. The distance between the foci is denoted by 2c, where c is the distance from the center of the hyperbola to either focus. The foci are equidistant from the center of the hyperbola, and their positions determine the shape and orientation of the hyperbola.

Identifying Foci in the Same Quadrant

To determine if the foci of a hyperbola lie in the same quadrant, we need to examine the equation of the hyperbola. The equation of a hyperbola in standard form is given by:

(x−h)2a2−(y−k)2b2=1\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1

where (h,k) is the center of the hyperbola, and a and b are the distances from the center to the vertices along the transverse and conjugate axes, respectively.

Analyzing the Given Options

Let's analyze the given options to determine which hyperbola has both foci lying in the same quadrant.

Option A: (x−24)2242−(y−1)272=1\frac{(x-24)^2}{24^2}-\frac{(y-1)^2}{7^2}=1

In this option, the x-term is positive, indicating that the transverse axis is horizontal. The center of the hyperbola is at (24,1), and the foci are located at (24 ± c,1). Since the x-coordinates of the foci are the same, the foci lie in the same quadrant.

Option B: (y−12)252−(x−6)2122=1\frac{(y-12)^2}{5^2}-\frac{(x-6)^2}{12^2}=1

In this option, the y-term is positive, indicating that the transverse axis is vertical. The center of the hyperbola is at (6,12), and the foci are located at (6,12 ± c). Since the y-coordinates of the foci are the same, the foci lie in the same quadrant.

Option C: (x−36)2362−(y−9)292=1\frac{(x-36)^2}{36^2}-\frac{(y-9)^2}{9^2}=1

In this option, the x-term is positive, indicating that the transverse axis is horizontal. The center of the hyperbola is at (36,9), and the foci are located at (36 ± c,9). Since the x-coordinates of the foci are different, the foci do not lie in the same quadrant.

Conclusion

Based on the analysis of the given options, we can conclude that both Option A and Option B have both foci lying in the same quadrant. However, only Option A has the foci lying in the same quadrant as the center of the hyperbola.

Key Takeaways

  • Foci are the two fixed points that define a hyperbola.
  • The distance between the foci is denoted by 2c, where c is the distance from the center of the hyperbola to either focus.
  • To determine if the foci of a hyperbola lie in the same quadrant, we need to examine the equation of the hyperbola.
  • The equation of a hyperbola in standard form is given by (x−h)2a2−(y−k)2b2=1\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1.
  • The foci of a hyperbola lie in the same quadrant if the x-coordinates or y-coordinates of the foci are the same.

Final Answer

The correct answer is Option A: (x−24)2242−(y−1)272=1\frac{(x-24)^2}{24^2}-\frac{(y-1)^2}{7^2}=1.

Introduction

Hyperbolas are a fundamental concept in mathematics, and understanding their properties is crucial for solving various problems in algebra, geometry, and calculus. One of the key aspects of hyperbolas is the location of their foci. In this article, we will delve into the world of hyperbola foci and provide a comprehensive Q&A guide to help you understand this concept better.

Q: What are foci in hyperbolas?

A: Foci are the two fixed points that define a hyperbola. They are located on the transverse axis, which is the axis that is perpendicular to the conjugate axis. The distance between the foci is denoted by 2c, where c is the distance from the center of the hyperbola to either focus.

Q: How do I determine the location of the foci of a hyperbola?

A: To determine the location of the foci of a hyperbola, you need to examine the equation of the hyperbola. The equation of a hyperbola in standard form is given by:

(x−h)2a2−(y−k)2b2=1\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1

where (h,k) is the center of the hyperbola, and a and b are the distances from the center to the vertices along the transverse and conjugate axes, respectively.

Q: What is the relationship between the foci and the center of a hyperbola?

A: The foci of a hyperbola are equidistant from the center of the hyperbola. The distance between the foci is denoted by 2c, where c is the distance from the center of the hyperbola to either focus.

Q: Can the foci of a hyperbola lie in the same quadrant?

A: Yes, the foci of a hyperbola can lie in the same quadrant. This occurs when the x-coordinates or y-coordinates of the foci are the same.

Q: How do I determine if the foci of a hyperbola lie in the same quadrant?

A: To determine if the foci of a hyperbola lie in the same quadrant, you need to examine the equation of the hyperbola. If the x-term is positive, the transverse axis is horizontal, and if the y-term is positive, the transverse axis is vertical.

Q: What is the significance of the foci of a hyperbola?

A: The foci of a hyperbola are significant because they determine the shape and orientation of the hyperbola. The distance between the foci is also important because it is related to the eccentricity of the hyperbola.

Q: Can the foci of a hyperbola be located at the center of the hyperbola?

A: No, the foci of a hyperbola cannot be located at the center of the hyperbola. The foci are always located on the transverse axis, which is the axis that is perpendicular to the conjugate axis.

Q: How do I find the distance between the foci of a hyperbola?

A: To find the distance between the foci of a hyperbola, you need to use the formula 2c, where c is the distance from the center of the hyperbola to either focus.

Q: What is the relationship between the foci and the vertices of a hyperbola?

A: The foci of a hyperbola are related to the vertices of the hyperbola. The distance between the foci is denoted by 2c, where c is the distance from the center of the hyperbola to either focus.

Conclusion

In conclusion, the foci of a hyperbola are an essential concept in mathematics. Understanding the location and properties of the foci is crucial for solving various problems in algebra, geometry, and calculus. We hope that this Q&A guide has provided you with a comprehensive understanding of the foci of a hyperbola.

Key Takeaways

  • Foci are the two fixed points that define a hyperbola.
  • The distance between the foci is denoted by 2c, where c is the distance from the center of the hyperbola to either focus.
  • The foci of a hyperbola can lie in the same quadrant.
  • The equation of a hyperbola in standard form is given by (x−h)2a2−(y−k)2b2=1\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1.
  • The foci of a hyperbola are related to the vertices of the hyperbola.

Final Answer

The foci of a hyperbola are the two fixed points that define the shape and orientation of the hyperbola. Understanding the location and properties of the foci is crucial for solving various problems in mathematics.