The Equation X 2 24 2 − Y 2 [ … ] 2 = 1 \frac{x^2}{24^2}-\frac{y^2}{[\ldots]^2}=1 2 4 2 X 2 ​ − [ … ] 2 Y 2 ​ = 1 Represents A Hyperbola Centered At The Origin With A Directrix Of X = 576 26 X=\frac{576}{26} X = 26 576 ​ .\begin{tabular}{|l|l|}\hline Vertices: ( − A , 0 ) , ( A , 0 (-a, 0),(a, 0 ( − A , 0 ) , ( A , 0 ] & Vertices: $(0,-a),(0,

A hyperbola is a type of mathematical curve 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 (called foci) is constant. In this article, we will explore the equation of a hyperbola, specifically the equation $\frac{x2}{242}-\frac{y2}{[\ldots]2}=1$, and understand the concept of vertices and directrix.

Understanding the Equation of a Hyperbola

The equation of a hyperbola can be written in the form $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, where $a$ and $b$ are the distances from the center to the vertices along the x-axis and y-axis, respectively. In the given equation, $\frac{x^2}{24^2}-\frac{y^2}{[\ldots]^2}=1$, we can see that $a=24$. However, the value of $b$ is not explicitly given.

Finding the Value of $b$

To find the value of $b$, we need to use the given information about the directrix. The directrix of a hyperbola is a line that is perpendicular to the transverse axis and is located at a distance of $c$ from the center, where $c^2=a^2+b^2$. In this case, the directrix is given as $x=\frac{576}{26}$. We can use this information to find the value of $c$.

Calculating the Value of $c$

To calculate the value of $c$, we need to find the distance from the center to the directrix. The distance from the center to the directrix is given by $c=\frac{576}{26}$. Now, we can use the equation $c^2=a^2+b^2$ to find the value of $b$.

Solving for $b$

We can rearrange the equation $c^2=a^2+b^2$ to solve for $b$. We get $b^2=c^2-a^2$. Substituting the values of $c$ and $a$, we get $b^2=\left(\frac{576}{26}\right)^2-24^2$. Simplifying, we get $b^2=\frac{331776}{676}-576$. Further simplifying, we get $b^2=\frac{331776-393216}{676}$. This gives us $b^2=\frac{-61640}{676}$. Taking the square root of both sides, we get $b=\sqrt{\frac{-61640}{676}}$. Simplifying, we get $b=\sqrt{-91}$.

Understanding the Vertices of a Hyperbola

The vertices of a hyperbola are the points on the transverse axis that are closest to the center. In the given equation, the vertices are given as $(-a, 0)$ and $(a, 0)$. Since $a=24$, the vertices are $(-24, 0)$ and $(24, 0)$.

Understanding the Directrix of a Hyperbola

The directrix of a hyperbola is a line that is perpendicular to the transverse axis and is located at a distance of $c$ from the center. In the given equation, the directrix is given as $x=\frac{576}{26}$. This means that the directrix is a vertical line that is located at a distance of $\frac{576}{26}$ from the center.

Conclusion

In this article, we have explored the equation of a hyperbola, specifically the equation $\frac{x^2}{24^2}-\frac{y^2}{[\ldots]^2}=1$. We have understood the concept of vertices and directrix and have calculated the value of $b$. We have also seen how to find the vertices and directrix of a hyperbola using the given equation.

Vertices of a Hyperbola

• Vertices: $(-a, 0)$ and $(a, 0)$
• Value of $a$: $24$
• Vertices: $(-24, 0)$ and $(24, 0)$

Directrix of a Hyperbola

• Directrix: $x=\frac{576}{26}$
• Distance from center to directrix: $\frac{576}{26}$

Calculating the Value of $b$

• Equation: $c^2=a^2+b^2$
• Value of $c$: $\frac{576}{26}$
• Value of $a$: $24$
• Value of $b$: $\sqrt{\frac{-61640}{676}}$

Simplifying the Value of $b$

• Value of $b$: $\sqrt{-91}$
  Frequently Asked Questions (FAQs) about Hyperbolas =====================================================

A hyperbola is a type of mathematical curve 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 (called 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 mathematical curve 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 (called foci) is constant.

Q: What is the equation of a hyperbola?

A: The equation of a hyperbola can be written in the form $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, where $a$ and $b$ are the distances from the center to the vertices along the x-axis and y-axis, respectively.

Q: What are the vertices of a hyperbola?

A: The vertices of a hyperbola are the points on the transverse axis that are closest to the center. In the given equation, the vertices are given as $(-a, 0)$ and $(a, 0)$.

Q: What is the directrix of a hyperbola?

A: The directrix of a hyperbola is a line that is perpendicular to the transverse axis and is located at a distance of $c$ from the center. In the given equation, the directrix is given as $x=\frac{576}{26}$.

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 given information about the directrix. The directrix is a line that is perpendicular to the transverse axis and is located at a distance of $c$ from the center. You can use the equation $c^2=a^2+b^2$ to find the value of $b$.

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

A: The value of $b$ represents the distance from the center to the vertices along the y-axis. It is an important parameter in the equation of a hyperbola and is used to determine the shape and size of the curve.

Q: Can you provide an example of how to find the value of $b$ in the equation of a hyperbola?

A: Let's consider the equation $\frac{x^2}{24^2}-\frac{y^2}{[\ldots]^2}=1$. We are given that the directrix is $x=\frac{576}{26}$. We can use the equation $c^2=a^2+b^2$ to find the value of $b$. Substituting the values of $c$ and $a$, we get $b^2=\left(\frac{576}{26}\right)^2-24^2$. Simplifying, we get $b^2=\frac{331776}{676}-576$. Further simplifying, we get $b^2=\frac{331776-393216}{676}$. This gives us $b^2=\frac{-61640}{676}$. Taking the square root of both sides, we get $b=\sqrt{\frac{-61640}{676}}$. Simplifying, we get $b=\sqrt{-91}$.

Q: What is the relationship between the value of $b$ and the shape of the hyperbola?

A: The value of $b$ determines the shape of the hyperbola. If $b$ is large, the hyperbola will be more elongated. If $b$ is small, the hyperbola will be more circular.

Q: Can you provide an example of how to use the value of $b$ to determine the shape of the hyperbola?

A: Let's consider the equation $\frac{x^2}{24^2}-\frac{y^2}{[\ldots]^2}=1$. We are given that the value of $b$ is $\sqrt{-91}$. Since $b$ is negative, the hyperbola will be more elongated.

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

A: The value of $a$ represents the distance from the center to the vertices along the x-axis. It is an important parameter in the equation of a hyperbola and is used to determine the shape and size of the curve.

Q: Can you provide an example of how to use the value of $a$ to determine the shape of the hyperbola?

A: Let's consider the equation $\frac{x^2}{24^2}-\frac{y^2}{[\ldots]^2}=1$. We are given that the value of $a$ is $24$. Since $a$ is positive, the hyperbola will be more circular.

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

A: The values of $a$ and $b$ are related by the equation $c^2=a^2+b^2$. This means that the value of $b$ is determined by the values of $a$ and $c$.

Q: Can you provide an example of how to use the values of $a$ and $b$ to determine the shape of the hyperbola?

A: Let's consider the equation $\frac{x^2}{24^2}-\frac{y^2}{[\ldots]^2}=1$. We are given that the values of $a$ and $b$ are $24$ and $\sqrt{-91}$, respectively. Since $b$ is negative, the hyperbola will be more elongated.