The Equation $\frac{(y+5)^2}{121}+\frac{(x-9)^2}{49}=1$ Represents An Ellipse.Which Point Is The Center Of The Ellipse?A. $(-9,5$\] B. $(-5,9$\] C. $(5,-9$\] D. $(9,-5$\]

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Introduction

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An ellipse is a fundamental concept in mathematics, and its equation is a crucial aspect of understanding this geometric shape. The equation of an ellipse in standard form is given by $\frac{(y-k)^2}{a^2} + \frac{(x-h)^2}{b^2} = 1$, where $(h,k)$ represents the center of the ellipse. In this article, we will focus on finding the center of an ellipse represented by the equation $\frac{(y+5)^2}{121} + \frac{(x-9)^2}{49} = 1$.

Understanding the Equation

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The given equation is in the standard form of an ellipse, where the center is represented by the point $(h,k)$. To find the center, we need to identify the values of $h$ and $k$ in the equation. In this case, the equation is $\frac{(y+5)^2}{121} + \frac{(x-9)^2}{49} = 1$. By comparing this equation with the standard form, we can see that $h = 9$ and $k = -5$.

Finding the Center

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Now that we have identified the values of $h$ and $k$, we can find the center of the ellipse. The center is represented by the point $(h,k)$, which in this case is $(9,-5)$. Therefore, the center of the ellipse is the point $(9,-5)$.

Conclusion

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In conclusion, the equation $\frac{(y+5)^2}{121} + \frac{(x-9)^2}{49} = 1$ represents an ellipse with a center at the point $(9,-5)$. This is a fundamental concept in mathematics, and understanding the equation of an ellipse is crucial for solving various mathematical problems.

Step-by-Step Solution

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Step 1: Identify the Standard Form of the Equation

The standard form of the equation of an ellipse is $\frac{(y-k)^2}{a^2} + \frac{(x-h)^2}{b^2} = 1$, where $(h,k)$ represents the center of the ellipse.

Step 2: Compare the Given Equation with the Standard Form

By comparing the given equation $\frac{(y+5)^2}{121} + \frac{(x-9)^2}{49} = 1$ with the standard form, we can see that $h = 9$ and $k = -5$.

Step 3: Find the Center of the Ellipse

The center of the ellipse is represented by the point $(h,k)$, which in this case is $(9,-5)$.

Frequently Asked Questions

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Q: What is the center of the ellipse represented by the equation $\frac{(y+5)^2}{121} + \frac{(x-9)^2}{49} = 1$?

A: The center of the ellipse is the point $(9,-5)$.

Q: How do I find the center of an ellipse represented by an equation in standard form?

A: To find the center of an ellipse represented by an equation in standard form, you need to identify the values of $h$ and $k$ in the equation.

Q: What is the significance of the center of an ellipse?

A: The center of an ellipse is a fundamental concept in mathematics, and understanding the equation of an ellipse is crucial for solving various mathematical problems.

Final Answer

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The final answer is $\boxed{(9,-5)}$.

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Introduction

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In our previous article, we discussed the equation of an ellipse and how to find the center of the ellipse. In this article, we will provide a Q&A section to help you better understand the concept of the equation of an ellipse and how to find the center.

Q&A

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Q: What is the equation of an ellipse?

A: The equation of an ellipse is a mathematical representation of the shape of an ellipse. It is typically written in the standard form as $\frac{(y-k)^2}{a^2} + \frac{(x-h)^2}{b^2} = 1$, where $(h,k)$ represents the center of the ellipse.

Q: What is the significance of the center of an ellipse?

A: The center of an ellipse is a fundamental concept in mathematics, and understanding the equation of an ellipse is crucial for solving various mathematical problems. The center of the ellipse is the point around which the ellipse is symmetric.

Q: How do I find the center of an ellipse represented by an equation in standard form?

A: To find the center of an ellipse represented by an equation in standard form, you need to identify the values of $h$ and $k$ in the equation. In the equation $\frac{(y+5)^2}{121} + \frac{(x-9)^2}{49} = 1$, the values of $h$ and $k$ are $h = 9$ and $k = -5$, respectively.

Q: What is the difference between the equation of an ellipse and the equation of a circle?

A: The equation of a circle is $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{a^2} = 1$, where $(h,k)$ represents the center of the circle. The equation of an ellipse is $\frac{(y-k)^2}{a^2} + \frac{(x-h)^2}{b^2} = 1$, where $(h,k)$ represents the center of the ellipse. The key difference is that the equation of an ellipse has two different values for $a$ and $b$, while the equation of a circle has the same value for $a$ and $b$.

Q: Can you provide an example of an ellipse with a center at the point $(0,0)$?

A: Yes, the equation of an ellipse with a center at the point $(0,0)$ is $\frac{y^2}{a^2} + \frac{x^2}{b^2} = 1$. For example, if $a = 5$ and $b = 3$, the equation of the ellipse is $\frac{y^2}{25} + \frac{x^2}{9} = 1$.

Q: How do I graph an ellipse represented by an equation in standard form?

A: To graph an ellipse represented by an equation in standard form, you need to identify the values of $h$ and $k$ in the equation. Then, you can plot the center of the ellipse at the point $(h,k)$. Next, you can plot the vertices of the ellipse, which are located at the points $(h \pm a, k)$ and $(h, k \pm b)$. Finally, you can plot the co-vertices of the ellipse, which are located at the points $(h \pm b, k)$ and $(h, k \pm a)$.

Conclusion

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In conclusion, the equation of an ellipse is a fundamental concept in mathematics, and understanding the equation of an ellipse is crucial for solving various mathematical problems. We hope that this Q&A article has helped you better understand the concept of the equation of an ellipse and how to find the center.

Step-by-Step Solution

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Step 1: Identify the Standard Form of the Equation

The standard form of the equation of an ellipse is $\frac{(y-k)^2}{a^2} + \frac{(x-h)^2}{b^2} = 1$, where $(h,k)$ represents the center of the ellipse.

Step 2: Compare the Given Equation with the Standard Form

By comparing the given equation with the standard form, you can identify the values of $h$ and $k$ in the equation.

Step 3: Find the Center of the Ellipse

The center of the ellipse is represented by the point $(h,k)$, which can be found by identifying the values of $h$ and $k$ in the equation.

Frequently Asked Questions

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Q: What is the equation of an ellipse?

A: The equation of an ellipse is a mathematical representation of the shape of an ellipse.

Q: What is the significance of the center of an ellipse?

A: The center of an ellipse is a fundamental concept in mathematics, and understanding the equation of an ellipse is crucial for solving various mathematical problems.

Q: How do I find the center of an ellipse represented by an equation in standard form?

A: To find the center of an ellipse represented by an equation in standard form, you need to identify the values of $h$ and $k$ in the equation.

Final Answer

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The final answer is $\boxed{(9,-5)}$.