The Equation Y 2 8 2 − X 2 B 2 \frac{y^2}{8^2}-\frac{x^2}{b^2} 8 2 Y 2 ​ − B 2 X 2 ​ Represents A Hyperbola Centered At The Origin With A Focus Of ( 0 , − 10 (0,-10 ( 0 , − 10 ]. What Is The Value Of B B B ?A. 4 B. 6 C. 10 D. 12

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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 (foci) is constant. In this article, we will explore the equation of a hyperbola centered at the origin with a focus of $(0,-10)$ and find the value of $b$.

The Equation of a Hyperbola

The standard equation of a hyperbola centered at the origin is given by:

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

where $a$ and $b$ are the lengths of the semi-major and semi-minor axes, respectively. In this case, the equation is:

$$\frac{y^2}{8^2}-\frac{x^2}{b^2}=1$$

Finding the Value of $b$

To find the value of $b$, we need to use the information given in the problem. We are told that the focus of the hyperbola is at $(0,-10)$. The distance from the center of the hyperbola to the focus is given by:

$$c=\sqrt{a^2+b^2}$$

In this case, $c=10$ and $a=8$. We can substitute these values into the equation above to get:

$$10=\sqrt{8^2+b^2}$$

Squaring both sides of the equation, we get:

$$100=64+b^2$$

Subtracting 64 from both sides, we get:

$$36=b^2$$

Taking the square root of both sides, we get:

$$b=\pm6$$

However, since $b$ represents the length of the semi-minor axis, it must be positive. Therefore, the value of $b$ is:

$$b=6$$

Conclusion

In this article, we have explored the equation of a hyperbola centered at the origin with a focus of $(0,-10)$. We have used the information given in the problem to find the value of $b$, which is the length of the semi-minor axis. The value of $b$ is 6.

Answer

The final answer is $\boxed{6}$.

References

• [1] "Hyperbola" by Math Open Reference. Retrieved 2023-12-01.
• [2] "Conic Sections" by Khan Academy. Retrieved 2023-12-01.

Related Topics

• Hyperbola
• Conic Sections
• Mathematics
  Q&A: The Equation of a Hyperbola =====================================

Introduction

In our previous article, we explored the equation of a hyperbola centered at the origin with a focus of $(0,-10)$. We found the value of $b$, which is the length of the semi-minor axis. In this article, we will answer some frequently asked questions related to the equation of a hyperbola.

Q: What is the standard equation of a hyperbola?

A: The standard equation of a hyperbola centered at the origin is given by:

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

Q: What is the difference between the semi-major axis and the semi-minor axis?

A: The semi-major axis is the length of the longest diameter of the hyperbola, while the semi-minor axis is the length of the shortest diameter of the hyperbola.

Q: How do I find the value of $b$ in the equation of a hyperbola?

A: To find the value of $b$, you need to use the information given in the problem. You can use the equation:

$$c=\sqrt{a^2+b^2}$$

where $c$ is the distance from the center of the hyperbola to the focus.

Q: What is the significance of the value of $b$ in the equation of a hyperbola?

A: The value of $b$ represents the length of the semi-minor axis of the hyperbola. It is an important parameter in the equation of a hyperbola.

Q: Can I use the equation of a hyperbola to find the value of $a$?

A: Yes, you can use the equation of a hyperbola to find the value of $a$. You can rearrange the equation:

$$c=\sqrt{a^2+b^2}$$

to solve for $a$.

Q: What is the relationship between the equation of a hyperbola and the conic sections?

A: The equation of a hyperbola is a type of conic section. Conic sections are a family of curves that include circles, ellipses, parabolas, and hyperbolas.

Q: Can I use the equation of a hyperbola to find the coordinates of the foci?

A: Yes, you can use the equation of a hyperbola to find the coordinates of the foci. The coordinates of the foci are given by:

$$(0, \pm c)$$

where $c$ is the distance from the center of the hyperbola to the focus.

Conclusion

In this article, we have answered some frequently asked questions related to the equation of a hyperbola. We have covered topics such as the standard equation of a hyperbola, the difference between the semi-major axis and the semi-minor axis, and the significance of the value of $b$ in the equation of a hyperbola.

References

• [1] "Hyperbola" by Math Open Reference. Retrieved 2023-12-01.
• [2] "Conic Sections" by Khan Academy. Retrieved 2023-12-01.

Related Topics

• Hyperbola
• Conic Sections
• Mathematics