What Are The Coordinates Of The Center Of This Hyperbola?$\frac{(y+3)^2}{25}-\frac{(x-4)^2}{36}=1$A. \[$(-3, 4)\$\]B. \[$(4, -3)\$\]C. \[$(3, -4)\$\]D. \[$(-4, 3)\$\]

Understanding 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. The standard form of a hyperbola is given by the equation:

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

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.

Identifying the Center of a Hyperbola

To identify the center of a hyperbola, we need to look at the equation and find the values of $h$ and $k$. In the given equation:

$\frac{(y+3)^2}{25}-\frac{(x-4)^2}{36}=1$

we can see that the value of $h$ is $4$ and the value of $k$ is $-3$. Therefore, the coordinates of the center of the hyperbola are $(4, -3)$.

Why is this the correct answer?

The equation of a hyperbola is given by the formula:

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

where $(h,k)$ is the center of the hyperbola. In the given equation, we can see that the value of $h$ is $4$ and the value of $k$ is $-3$. Therefore, the coordinates of the center of the hyperbola are $(4, -3)$.

Conclusion

In conclusion, the coordinates of the center of the hyperbola are $(4, -3)$. This is because the value of $h$ is $4$ and the value of $k$ is $-3$ in the given equation.

Answer

The correct answer is:

B. {(4, -3)$}$

Why is this the correct answer?

The correct answer is B. {(4, -3)$}$ because the value of $h$ is $4$ and the value of $k$ is $-3$ in the given equation.

What are the other options?

The other options are:

A. {(-3, 4)$}$

C. {(3, -4)$}$

D. {(-4, 3)$}$

These options are incorrect because the value of $h$ is not $-3$, $3$, or $-4$, and the value of $k$ is not $4$, $-4$, or $3$ in the given equation.

Why are these options incorrect?

These options are incorrect because the value of $h$ is not $-3$, $3$, or $-4$, and the value of $k$ is not $4$, $-4$, or $3$ in the given equation. The correct values of $h$ and $k$ are $4$ and $-3$, respectively.

What is the significance of the center of a hyperbola?

The center of a hyperbola is significant because it is the point around which the hyperbola is symmetric. The center is also the point where the two branches of the hyperbola intersect. In addition, the center is used to determine the orientation of the hyperbola, which is either horizontal or vertical.

How is the center of a hyperbola used in real-world applications?

The center of a hyperbola is used in real-world applications such as:

• Astronomy: The center of a hyperbola is used to determine the trajectory of a comet or a planet.
• Physics: The center of a hyperbola is used to describe the motion of an object under the influence of a central force.
• Engineering: The center of a hyperbola is used to design and optimize the shape of a hyperbolic reflector or a hyperbolic lens.

Conclusion

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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 standard form of a hyperbola?

A: The standard form of a hyperbola is given by the equation:

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

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: How do I identify the center of a hyperbola?

A: To identify the center of a hyperbola, we need to look at the equation and find the values of $h$ and $k$. In the given equation:

$\frac{(y+3)^2}{25}-\frac{(x-4)^2}{36}=1$

we can see that the value of $h$ is $4$ and the value of $k$ is $-3$. Therefore, the coordinates of the center of the hyperbola are $(4, -3)$.

Q: Why is this the correct answer?

A: The equation of a hyperbola is given by the formula:

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

where $(h,k)$ is the center of the hyperbola. In the given equation, we can see that the value of $h$ is $4$ and the value of $k$ is $-3$. Therefore, the coordinates of the center of the hyperbola are $(4, -3)$.

Q: What are the other options?

A: The other options are:

A. {(-3, 4)$}$

C. {(3, -4)$}$

D. {(-4, 3)$}$

Q: Why are these options incorrect?

A: These options are incorrect because the value of $h$ is not $-3$, $3$, or $-4$, and the value of $k$ is not $4$, $-4$, or $3$ in the given equation. The correct values of $h$ and $k$ are $4$ and $-3$, respectively.

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

A: The center of a hyperbola is significant because it is the point around which the hyperbola is symmetric. The center is also the point where the two branches of the hyperbola intersect. In addition, the center is used to determine the orientation of the hyperbola, which is either horizontal or vertical.

Q: How is the center of a hyperbola used in real-world applications?

A: The center of a hyperbola is used in real-world applications such as:

• Astronomy: The center of a hyperbola is used to determine the trajectory of a comet or a planet.
• Physics: The center of a hyperbola is used to describe the motion of an object under the influence of a central force.
• Engineering: The center of a hyperbola is used to design and optimize the shape of a hyperbolic reflector or a hyperbolic lens.

Q: What are some common mistakes to avoid when working with hyperbolas?

A: Some common mistakes to avoid when working with hyperbolas include:

• Not identifying the center of the hyperbola correctly: Make sure to identify the values of $h$ and $k$ correctly in the equation.
• Not understanding the orientation of the hyperbola: Make sure to understand whether the hyperbola is horizontal or vertical.
• Not using the correct formula for the equation of a hyperbola: Make sure to use the correct formula for the equation of a hyperbola.

Q: How can I practice working with hyperbolas?

A: You can practice working with hyperbolas by:

• Solving problems and exercises: Practice solving problems and exercises that involve hyperbolas.
• Graphing hyperbolas: Practice graphing hyperbolas to visualize their shape and orientation.
• Using online resources: Use online resources such as calculators and graphing tools to help you practice working with hyperbolas.