The Equation Y 2 8 2 − X 2 B 2 = 1 \frac{y^2}{8^2} - \frac{x^2}{b^2} = 1 8 2 Y 2 ​ − B 2 X 2 ​ = 1 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. 6 B. 12 C. 4 D. 10

Introduction

Hyperbolas are 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 equation of a hyperbola can be written in the form $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$, where $(0, a)$ is the center of the hyperbola and the foci are located at $(0, \pm c)$. In this problem, we are given the equation $\frac{y^2}{8^2} - \frac{x^2}{b^2} = 1$ and we need to find the value of $b$.

Understanding the Equation of a Hyperbola

The equation of a hyperbola can be written in the form $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$. In this equation, $a$ is the distance from the center of the hyperbola to the vertices, and $b$ is the distance from the center of the hyperbola to the co-vertices. The foci of the hyperbola are located at $(0, \pm c)$, where $c^2 = a^2 + b^2$. In this problem, we are given that the foci are located at $(0, -10)$, which means that $c = 10$.

Finding the Value of b

We are given the equation $\frac{y^2}{8^2} - \frac{x^2}{b^2} = 1$. Since the foci are located at $(0, -10)$, we know that $c = 10$. We can use the equation $c^2 = a^2 + b^2$ to find the value of $b$. Plugging in the values, we get:

$$10^2 = 8^2 + b^2$$

Simplifying 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:

$$b = \pm 6$$

However, since $b$ is a distance, it must be positive. Therefore, the value of $b$ is 6.

Conclusion

In this problem, we were given the equation $\frac{y^2}{8^2} - \frac{x^2}{b^2} = 1$ and we needed to find the value of $b$. We used the equation $c^2 = a^2 + b^2$ to find the value of $b$, and we found that $b = 6$. Therefore, the correct answer is A. 6.

Key Takeaways

• The equation of a hyperbola can be written in the form $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$.
• The foci of a hyperbola are located at $(0, \pm c)$, where $c^2 = a^2 + b^2$.
• The value of $b$ can be found using the equation $c^2 = a^2 + b^2$.

Practice Problems

• Find the value of $b$ for the equation $\frac{y^2}{12^2} - \frac{x^2}{b^2} = 1$, given that the foci are located at $(0, -15)$.
• Find the value of $a$ for the equation $\frac{y^2}{a^2} - \frac{x^2}{16^2} = 1$, given that the foci are located at $(0, -20)$.

Solutions

• For the first problem, we can use the equation $c^2 = a^2 + b^2$ to find the value of $b$. Plugging in the values, we get:

$$15^2 = 12^2 + b^2$$

Simplifying the equation, we get:

$$225 = 144 + b^2$$

Subtracting 144 from both sides, we get:

$$81 = b^2$$

Taking the square root of both sides, we get:

$$b = \pm 9$$

However, since $b$ is a distance, it must be positive. Therefore, the value of $b$ is 9.

• For the second problem, we can use the equation $c^2 = a^2 + b^2$ to find the value of $a$. Plugging in the values, we get:

$$20^2 = a^2 + 16^2$$

Simplifying the equation, we get:

$$400 = a^2 + 256$$

Subtracting 256 from both sides, we get:

$$144 = a^2$$

Taking the square root of both sides, we get:

$$a = \pm 12$$

However, since $a$ is a distance, it must be positive. Therefore, the value of $a$ is 12.

Introduction

Hyperbolas are 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 equation of a hyperbola can be written in the form $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$, where $(0, a)$ is the center of the hyperbola and the foci are located at $(0, \pm c)$. In this article, we will answer some common questions about hyperbolas and provide examples to help you understand the concept.

Q&A

Q: What is the equation of a hyperbola?

A: The equation of a hyperbola can be written in the form $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$, where $(0, a)$ is the center of the hyperbola and the foci are located at $(0, \pm c)$.

Q: What is the significance of the values a and b in the equation of a hyperbola?

A: The values $a$ and $b$ represent the distances from the center of the hyperbola to the vertices and co-vertices, respectively. The value $c$ represents the distance from the center of the hyperbola to the foci.

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

A: To find the value of $b$, you can use the equation $c^2 = a^2 + b^2$. Plugging in the values, you can solve for $b$.

Q: What is the relationship between the values a, b, and c in the equation of a hyperbola?

A: The values $a$, $b$, and $c$ are related by the equation $c^2 = a^2 + b^2$. This equation shows that the square of the distance from the center of the hyperbola to the foci is equal to the sum of the squares of the distances from the center of the hyperbola to the vertices and co-vertices.

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

A: To determine the center and foci of a hyperbola, you can use the equation of the hyperbola. The center of the hyperbola is the point $(0, a)$, and the foci are located at $(0, \pm c)$.

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

A: The value $c$ represents the distance from the center of the hyperbola to the foci. This value is used to determine the shape and size of the hyperbola.

Examples

Example 1: Finding the Value of b

Find the value of $b$ for the equation $\frac{y^2}{8^2} - \frac{x^2}{b^2} = 1$, given that the foci are located at $(0, -10)$.

Solution:

We can use the equation $c^2 = a^2 + b^2$ to find the value of $b$. Plugging in the values, we get:

$$10^2 = 8^2 + b^2$$

Simplifying 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:

$$b = \pm 6$$

However, since $b$ is a distance, it must be positive. Therefore, the value of $b$ is 6.

Example 2: Determining the Center and Foci of a Hyperbola

Determine the center and foci of the hyperbola $\frac{y^2}{12^2} - \frac{x^2}{16^2} = 1$.

Solution:

The center of the hyperbola is the point $(0, a)$, where $a$ is the value of $a$ in the equation of the hyperbola. In this case, $a = 12$. Therefore, the center of the hyperbola is the point $(0, 12)$.

The foci of the hyperbola are located at $(0, \pm c)$, where $c$ is the value of $c$ in the equation of the hyperbola. In this case, $c = \sqrt{a^2 + b^2} = \sqrt{12^2 + 16^2} = \sqrt{400} = 20$. Therefore, the foci of the hyperbola are located at $(0, \pm 20)$.

Conclusion

In this article, we have answered some common questions about hyperbolas and provided examples to help you understand the concept. We have discussed the equation of a hyperbola, the significance of the values $a$ and $b$, and how to find the value of $b$. We have also determined the center and foci of a hyperbola and discussed the significance of the value $c$. We hope that this article has been helpful in understanding the concept of hyperbolas.