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

Introduction

In mathematics, an ellipse is a fundamental concept in geometry and algebra. It is a closed curve on a plane surrounding two focal points such that the sum of the distances to the two focal points is constant. The equation of an ellipse can be represented in various forms, including the standard form, which is used to identify the center, vertices, and foci of the ellipse. In this article, we will focus on the equation $\frac{(y+5)^2}{121}+\frac{(x-9)^2}{49}=1$ and determine the center of the ellipse.

Understanding the Equation of an Ellipse

The standard form of the equation of an ellipse 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, and $a$ and $b$ are the lengths of the semi-major and semi-minor axes, respectively.

Analyzing the Given Equation

The given equation is:

$$\frac{(y+5)^2}{121}+\frac{(x-9)^2}{49}=1$$

Comparing this equation with the standard form, we can identify the values of $h$, $k$, $a$, and $b$.

• $h$ is the value that is being subtracted from $x$, which is $9$.
• $k$ is the value that is being added to $y$, which is $-5$.
• $a^2$ is the denominator of the first term, which is $121$.
• $b^2$ is the denominator of the second term, which is $49$.

Determining the Center of the Ellipse

To find the center of the ellipse, we need to identify the values of $h$ and $k$. From the given equation, we can see that $h=9$ and $k=-5$. Therefore, the center of the ellipse is at the point $(9, -5)$.

Conclusion

In conclusion, the equation $\frac{(y+5)^2}{121}+\frac{(x-9)^2}{49}=1$ represents an ellipse with its center at the point $(9, -5)$. This is determined by analyzing the given equation and comparing it with the standard form of the equation of an ellipse.

Answer

The correct answer is:

• C. $(9, -5)$

Discussion

This problem requires a good understanding of the equation of an ellipse and its standard form. The student needs to be able to identify the values of $h$, $k$, $a$, and $b$ from the given equation and use this information to determine the center of the ellipse. This problem is a good example of how to apply mathematical concepts to real-world problems and how to use algebraic techniques to solve problems in geometry.

Additional Examples

Here are a few more examples of how to find the center of an ellipse using the standard form of the equation:

• $\frac{(y-2)^2}{16}+\frac{(x+3)^2}{9}=1$
• $\frac{(y+1)^2}{25}+\frac{(x-4)^2}{4}=1$
• $\frac{(y-3)^2}{9}+\frac{(x+2)^2}{16}=1$

In each of these examples, the student needs to identify the values of $h$ and $k$ from the given equation and use this information to determine the center of the ellipse.

Tips and Tricks

Here are a few tips and tricks that can help students solve problems like this:

• Make sure to read the problem carefully and understand what is being asked.
• Use the standard form of the equation of an ellipse to identify the values of $h$, $k$, $a$, and $b$.
• Use algebraic techniques to simplify the equation and identify the center of the ellipse.
• Check your work by plugging the values of $h$ and $k$ back into the equation to make sure they are correct.

Introduction

In our previous article, we discussed the equation of an ellipse and how to find the center of the ellipse using the standard form of the equation. In this article, we will provide a Q&A section to help students better understand the concept of the equation of an ellipse and how to apply it to solve problems.

Q: What is the equation of an ellipse?

A: The equation of an ellipse is a mathematical representation of an ellipse in the form of a quadratic equation. It is used to describe the shape and size of an ellipse.

Q: What is the standard form of the equation of an ellipse?

A: The standard form of the equation of an ellipse 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, and $a$ and $b$ are the lengths of the semi-major and semi-minor axes, respectively.

Q: How do I find the center of the ellipse using the standard form of the equation?

A: To find the center of the ellipse, you need to identify the values of $h$ and $k$ from the given equation. $h$ is the value that is being subtracted from $x$, and $k$ is the value that is being added to $y$.

Q: What are the lengths of the semi-major and semi-minor axes?

A: The lengths of the semi-major and semi-minor axes are represented by $a$ and $b$, respectively. $a$ is the distance from the center of the ellipse to the vertex of the ellipse along the major axis, and $b$ is the distance from the center of the ellipse to the vertex of the ellipse along the minor axis.

Q: How do I determine the lengths of the semi-major and semi-minor axes?

A: To determine the lengths of the semi-major and semi-minor axes, you need to identify the values of $a^2$ and $b^2$ from the given equation. $a^2$ is the denominator of the first term, and $b^2$ is the denominator of the second term.

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

A: The semi-major axis is the longer axis of the ellipse, and the semi-minor axis is the shorter axis of the ellipse.

Q: How do I use the equation of an ellipse to solve problems?

A: To use the equation of an ellipse to solve problems, you need to identify the values of $h$, $k$, $a$, and $b$ from the given equation and use this information to determine the center, vertices, and foci of the ellipse.

Q: What are some common applications of the equation of an ellipse?

A: The equation of an ellipse has many common applications in mathematics, physics, and engineering, including:

• Describing the shape and size of an ellipse
• Finding the center, vertices, and foci of an ellipse
• Determining the lengths of the semi-major and semi-minor axes
• Solving problems involving ellipses in mathematics and physics

Conclusion

In conclusion, the equation of an ellipse is a fundamental concept in mathematics and has many common applications in mathematics, physics, and engineering. By understanding the standard form of the equation and how to apply it to solve problems, students can develop the skills and strategies they need to succeed in mathematics and other fields.

Additional Resources

For additional resources on the equation of an ellipse, including practice problems and interactive simulations, please visit the following websites:

• Khan Academy: Ellipse
• Mathway: Ellipse
• Wolfram Alpha: Ellipse

By using these resources, students can gain a deeper understanding of the equation of an ellipse and how to apply it to solve problems.