The Equation ( X − 7 ) 2 64 + ( Y + 2 ) 2 9 = 1 \frac{(x-7)^2}{64}+\frac{(y+2)^2}{9}=1 64 ( X − 7 ) 2 + 9 ( Y + 2 ) 2 = 1 Represents An Ellipse.Which Points Are The Vertices Of The Ellipse?- (7, -10) And (7, 6)- (10, -2) And (4, -2)- (15, -2) And (-1, -2)

Mar 9, 2025 by ADMIN 260 views

**The Equation of an Ellipse: Finding the Vertices**

Introduction

An ellipse is a fundamental concept in mathematics, and its equation is a crucial tool for understanding and analyzing its properties. In this article, we will focus on the equation of an ellipse in standard form, specifically the equation $\frac{(x-7)^2}{64}+\frac{(y+2)^2}{9}=1$ . Our goal is to identify the vertices of this ellipse, which are the points on the ellipse that lie on the major axis.

Understanding the Equation of an Ellipse

The equation of an ellipse in standard form is given by:

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

where $(h,k)$ is the center of the ellipse, and $a$ and $b$ are the lengths of the semi-major and semi-minor axes, respectively.

In our given equation, $\frac{(x-7)^2}{64}+\frac{(y+2)^2}{9}=1$ , we can identify the center of the ellipse as $(7,-2)$ , and the lengths of the semi-major and semi-minor axes as $8$ and $3$ , respectively.

Finding the Vertices of the Ellipse

The vertices of an ellipse are the points on the ellipse that lie on the major axis. Since the major axis is horizontal in this case, the vertices will have the same $y$ -coordinate as the center of the ellipse.

To find the vertices, we need to find the points on the ellipse that have the same $y$ -coordinate as the center, which is $-2$ . We can do this by substituting $y=-2$ into the equation of the ellipse and solving for $x$ .

Substituting $y=-2$ into the equation, we get:

\frac{(x-7)^2}{64}+\frac{(-2+2)^2}{9}=1

Simplifying the equation, we get:

\frac{(x-7)^2}{64}=1

Multiplying both sides by $64$ , we get:

(x-7)^2=64

Taking the square root of both sides, we get:

x-7=\pm8

Solving for $x$ , we get:

x=7\pm8

Therefore, the vertices of the ellipse are the points $(7+8,-2)$ and $(7-8,-2)$ , which are $(15,-2)$ and $(-1,-2)$ , respectively.

Conclusion

In this article, we have identified the vertices of the ellipse represented by the equation $\frac{(x-7)^2}{64}+\frac{(y+2)^2}{9}=1$ . We have shown that the vertices are the points $(15,-2)$ and $(-1,-2)$ , which lie on the major axis of the ellipse.

Discussion

The equation of an ellipse is a fundamental concept in mathematics, and its properties are crucial for understanding and analyzing various mathematical and real-world phenomena. The vertices of an ellipse are an essential part of its properties, and identifying them is a crucial step in understanding the ellipse's behavior.

In this article, we have focused on the equation of an ellipse in standard form and identified the vertices of the ellipse. We have also discussed the importance of understanding the equation of an ellipse and its properties.

Additional Resources

For further reading on the equation of an ellipse and its properties, we recommend the following resources:

Final Thoughts

Introduction

In our previous article, we explored the equation of an ellipse in standard form and identified the vertices of the ellipse represented by the equation $\frac{(x-7)^2}{64}+\frac{(y+2)^2}{9}=1$ . In this article, we will answer some frequently asked questions about the equation of an ellipse and its properties.

Q&A

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

A: The equation of an ellipse in standard form is given by:

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

where $(h,k)$ is the center of the ellipse, and $a$ and $b$ are the lengths of the semi-major and semi-minor axes, respectively.

Q: What are the vertices of an ellipse?

A: The vertices of an ellipse are the points on the ellipse that lie on the major axis. Since the major axis is horizontal in this case, the vertices will have the same $y$ -coordinate as the center of the ellipse.

Q: How do I find the vertices of an ellipse?

A: To find the vertices of an ellipse, you need to substitute the $y$ -coordinate of the center into the equation of the ellipse and solve for $x$ . This will give you the $x$ -coordinates of the vertices.

Q: What is the center of an ellipse?

A: The center of an ellipse is the point $(h,k)$ in the equation of the ellipse. This is the point around which the ellipse is centered.

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

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

Q: How do I determine the major and minor axes of an ellipse?

A: To determine the major and minor axes of an ellipse, you need to look at the coefficients of the $x^2$ and $y^2$ terms in the equation of the ellipse. The larger coefficient corresponds to the major axis, and the smaller coefficient corresponds to the minor axis.

Q: What is the foci of an ellipse?

A: The foci of an ellipse are the points inside the ellipse that are equidistant from the center of the ellipse. The foci are located on the major axis of the ellipse.

Q: How do I find the foci of an ellipse?

A: To find the foci of an ellipse, you need to use the formula:

c=\sqrt{a^2-b^2}

where $c$ is the distance from the center of the ellipse to the foci.

Q: What is the eccentricity of an ellipse?

A: The eccentricity of an ellipse is a measure of how elliptical the ellipse is. It is given by the formula:

e=\frac{c}{a}

where $e$ is the eccentricity, $c$ is the distance from the center of the ellipse to the foci, and $a$ is the length of the semi-major axis.

Q: How do I find the eccentricity of an ellipse?

A: To find the eccentricity of an ellipse, you need to use the formula:

e=\frac{c}{a}

where $c$ is the distance from the center of the ellipse to the foci, and $a$ is the length of the semi-major axis.

Conclusion

In this article, we have answered some frequently asked questions about the equation of an ellipse and its properties. We hope that this article has provided a clear and concise explanation of the equation of an ellipse and its properties.

Additional Resources

For further reading on the equation of an ellipse and its properties, we recommend the following resources:

Final Thoughts

The equation of an ellipse is a fundamental concept in mathematics, and its properties are crucial for understanding and analyzing various mathematical and real-world phenomena. We hope that this article has provided a clear and concise explanation of the equation of an ellipse and its properties. We also hope that this article has inspired readers to explore the fascinating world of mathematics and its applications.