Which Is A True Statement Comparing The Graphs Of X 2 6 2 − Y 2 8 2 = 1 \frac{x^2}{6^2}-\frac{y^2}{8^2}=1 6 2 X 2 ​ − 8 2 Y 2 ​ = 1 And X 2 8 2 − Y 2 6 2 = 1 \frac{x^2}{8^2}-\frac{y^2}{6^2}=1 8 2 X 2 ​ − 6 2 Y 2 ​ = 1 ?A. The Lengths Of Both Transverse Axes Are The Same.B. The Directrices Of

Introduction

Hyperbolas are a fundamental concept in mathematics, particularly in the field of algebra and geometry. They are defined as the set of all points such that the absolute value of the difference between the distances from two fixed points (foci) is constant. In this article, we will compare the graphs of two hyperbolas, $\frac{x^2}{6^2}-\frac{y^2}{8^2}=1$ and $\frac{x^2}{8^2}-\frac{y^2}{6^2}=1$, and determine which statement is true.

Understanding Hyperbolas

A hyperbola is a type of conic section that can be defined as the set of all points such that the absolute value of the difference between the distances from two fixed points (foci) is constant. 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 the lengths of the transverse and conjugate axes, respectively.

Graphing Hyperbolas

To graph a hyperbola, we need to identify the center, vertices, and asymptotes. The center of a hyperbola is the point that is equidistant from the two foci. The vertices are the points on the hyperbola that are closest to the center. The asymptotes are the lines that the hyperbola approaches as it extends towards infinity.

Comparing the Graphs

Let's compare the graphs of the two hyperbolas, $\frac{x^2}{6^2}-\frac{y^2}{8^2}=1$ and $\frac{x^2}{8^2}-\frac{y^2}{6^2}=1$. We can see that the only difference between the two equations is the order of the terms. In the first equation, the $x^2$ term is positive, while in the second equation, the $y^2$ term is positive.

Analyzing the Transverse Axes

The transverse axis of a hyperbola is the axis that is perpendicular to the conjugate axis. The length of the transverse axis is given by $2a$. In the first equation, the length of the transverse axis is $2 \times 6 = 12$. In the second equation, the length of the transverse axis is $2 \times 8 = 16$.

Analyzing the Directrices

The directrices of a hyperbola are the lines that are perpendicular to the transverse axis and pass through the foci. The equation of the directrices is given by $x = \pm \frac{a^2}{c}$, where $c$ is the distance from the center to the foci. In the first equation, the directrices are given by $x = \pm \frac{6^2}{\sqrt{6^2+8^2}} = \pm \frac{36}{10}$. In the second equation, the directrices are given by $x = \pm \frac{8^2}{\sqrt{8^2+6^2}} = \pm \frac{64}{10}$.

Conclusion

Based on our analysis, we can conclude that the lengths of the transverse axes are not the same. The length of the transverse axis of the first hyperbola is 12, while the length of the transverse axis of the second hyperbola is 16. Therefore, statement A is false.

However, we can also conclude that the directrices of the two hyperbolas are not the same. The directrices of the first hyperbola are given by $x = \pm \frac{36}{10}$, while the directrices of the second hyperbola are given by $x = \pm \frac{64}{10}$. Therefore, statement B is also false.

Final Answer

The final answer is that neither statement A nor statement B is true. The lengths of the transverse axes are not the same, and the directrices of the two hyperbolas are not the same.

References

• [1] "Hyperbolas" by Math Open Reference
• [2] "Conic Sections" by Khan Academy
• [3] "Hyperbola" by Wolfram MathWorld

Additional Resources

• [1] "Hyperbolas" by Math Is Fun
• [2] "Conic Sections" by Purplemath
• [3] "Hyperbola" by IXL Math
  

Introduction

Hyperbolas are a fundamental concept in mathematics, particularly in the field of algebra and geometry. They are defined as the set of all points such that the absolute value of the difference between the distances from two fixed points (foci) is constant. 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 can be defined as the set of all points such that the absolute value of the difference between the distances from two fixed points (foci) is constant.

Q: What is the general equation of a hyperbola?

A: 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 the lengths of the transverse and conjugate axes, respectively.

Q: What are the foci of a hyperbola?

A: The foci of a hyperbola are the two fixed points that are equidistant from the center of the hyperbola. The distance from the center to the foci is given by $c = \sqrt{a^2+b^2}$.

Q: What are the vertices of a hyperbola?

A: The vertices of a hyperbola are the two points on the hyperbola that are closest to the center. The vertices are given by $(\pm a, 0)$.

Q: What are the asymptotes of a hyperbola?

A: The asymptotes of a hyperbola are the two lines that the hyperbola approaches as it extends towards infinity. The asymptotes are given by $y = \pm \frac{b}{a}x$.

Q: How do I graph a hyperbola?

A: To graph a hyperbola, you need to identify the center, vertices, and asymptotes. The center is the point that is equidistant from the two foci. The vertices are the points on the hyperbola that are closest to the center. The asymptotes are the lines that the hyperbola approaches as it extends towards infinity.

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

A: A hyperbola and an ellipse are both conic sections, but they have different properties. A hyperbola is defined as the set of all points such that the absolute value of the difference between the distances from two fixed points (foci) is constant. An ellipse is defined as the set of all points such that the sum of the distances from two fixed points (foci) is constant.

Q: Can a hyperbola have a negative value?

A: Yes, a hyperbola can have a negative value. In fact, the equation of a hyperbola can be written as $\frac{x^2}{a^2}-\frac{y^2}{b^2}=-1$, where $a$ and $b$ are the lengths of the transverse and conjugate axes, respectively.

Q: Can a hyperbola have a zero value?

A: Yes, a hyperbola can have a zero value. In fact, the equation of a hyperbola can be written as $\frac{x^2}{a^2}-\frac{y^2}{b^2}=0$, where $a$ and $b$ are the lengths of the transverse and conjugate axes, respectively.

Conclusion

In conclusion, hyperbolas are a fundamental concept in mathematics, particularly in the field of algebra and geometry. They are defined as the set of all points such that the absolute value of the difference between the distances from two fixed points (foci) is constant. We hope that this article has helped to answer some of the frequently asked questions about hyperbolas.

References

• [1] "Hyperbolas" by Math Open Reference
• [2] "Conic Sections" by Khan Academy
• [3] "Hyperbola" by Wolfram MathWorld

Additional Resources

• [1] "Hyperbolas" by Math Is Fun
• [2] "Conic Sections" by Purplemath
• [3] "Hyperbola" by IXL Math