The Equation $\frac{(x-7)^2}{64} + \frac{(y+2)^2}{9} = 1$ Represents An Ellipse. Which Points Are The Vertices Of The Ellipse?A. (7, 1) And (7, -5)B. (7, -10) And (7, 6)C. (10, -2) And (4, -2)D. (15, -2) And (-1, -2)

Feb 28, 2025 by ADMIN 217 views

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

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Introduction

In mathematics, an ellipse is a fundamental concept that is used to describe various shapes and patterns in geometry and trigonometry. The equation of an ellipse is given by the formula $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$ , 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 this article, we will focus on finding the vertices of an ellipse represented by the equation $\frac{(x-7)^2}{64} + \frac{(y+2)^2}{9} = 1$ .

Understanding the Equation

To find the vertices of the ellipse, we need to understand the equation and its components. The equation is in the standard form of an ellipse, which is $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$ . In this equation, $(h,k)$ is the center of the ellipse, which is $(7,-2)$ in this case. The values of $a$ and $b$ represent the lengths of the semi-major and semi-minor axes, respectively. In this equation, $a = 8$ and $b = 3$ .

Finding the Vertices

The vertices of an ellipse are the points on the ellipse that are farthest from the center. In the case of an ellipse with a horizontal major axis, the vertices are located at $(h+a,k)$ and $(h-a,k)$ . Similarly, for an ellipse with a vertical major axis, the vertices are located at $(h,k+b)$ and $(h,k-b)$ . In this case, the major axis is horizontal, so the vertices are located at $(7+8,-2)$ and $(7-8,-2)$ .

Calculating the Vertices

To calculate the vertices, we need to substitute the values of $h$ , $a$ , and $k$ into the formulas for the vertices. The vertices are located at $(7+8,-2)$ and $(7-8,-2)$ . Therefore, the vertices are $(15,-2)$ and $(-1,-2)$ .

Conclusion

In conclusion, the vertices of the ellipse represented by the equation $\frac{(x-7)^2}{64} + \frac{(y+2)^2}{9} = 1$ are $(15,-2)$ and $(-1,-2)$ . These points are the farthest points from the center of the ellipse and are located on the major axis of the ellipse.

Answer

The correct answer is:

D. (15, -2) and (-1, -2)

Discussion

This problem requires a good understanding of the equation of an ellipse and its components. The student needs to be able to identify the center, semi-major and semi-minor axes, and the vertices of the ellipse. The student also needs to be able to calculate the vertices using the formulas for the vertices.

Tips and Tricks

Make sure to understand the equation of an ellipse and its components.
Identify the center, semi-major and semi-minor axes, and the vertices of the ellipse.
Use the formulas for the vertices to calculate the vertices.
Check your answer by plugging the vertices back into the equation of the ellipse.

References

[1] "Ellipses" by Math Open Reference. Retrieved February 2023.
[2] "Equation of an Ellipse" by Purplemath. Retrieved February 2023.

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Q: What is an ellipse?

A: An ellipse 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. It is a fundamental concept in mathematics and is used to describe various shapes and patterns in geometry and trigonometry.

Q: What is the equation of an ellipse?

A: The equation of an ellipse is given by the formula $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$ , 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 are farthest from the center. In the case of an ellipse with a horizontal major axis, the vertices are located at $(h+a,k)$ and $(h-a,k)$ . Similarly, for an ellipse with a vertical major axis, the vertices are located at $(h,k+b)$ and $(h,k-b)$ .

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

A: To find the vertices of an ellipse, you need to identify the center, semi-major and semi-minor axes, and then use the formulas for the vertices. For an ellipse with a horizontal major axis, the vertices are located at $(h+a,k)$ and $(h-a,k)$ . For an ellipse with a vertical major axis, the vertices are located at $(h,k+b)$ and $(h,k-b)$ .

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

A: A circle is a special type of ellipse where the semi-major and semi-minor axes are equal. In other words, a circle is an ellipse with $a = b$ . An ellipse, on the other hand, 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.

Q: Can an ellipse have a vertical major axis?

A: Yes, an ellipse can have a vertical major axis. In this case, the vertices are located at $(h,k+b)$ and $(h,k-b)$ .

Q: How do I graph an ellipse?

A: To graph an ellipse, you need to identify the center, semi-major and semi-minor axes, and then plot the vertices and the major and minor axes. You can use a graphing calculator or a computer program to graph an ellipse.

Q: What is the equation of an ellipse with a horizontal major axis and a center at (3,4) and a semi-major axis of length 6 and a semi-minor axis of length 3?

A: The equation of an ellipse with a horizontal major axis and a center at $(3,4)$ and a semi-major axis of length $6$ and a semi-minor axis of length $3$ is $\frac{(x-3)^2}{36} + \frac{(y-4)^2}{9} = 1$ .

Q: What is the equation of an ellipse with a vertical major axis and a center at (2,5) and a semi-major axis of length 4 and a semi-minor axis of length 2?

A: The equation of an ellipse with a vertical major axis and a center at $(2,5)$ and a semi-major axis of length $4$ and a semi-minor axis of length $2$ is $\frac{(x-2)^2}{4} + \frac{(y-5)^2}{16} = 1$ .

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

A: To find the area of an ellipse, you need to use the formula $A = \pi ab$ , where $a$ and $b$ are the lengths of the semi-major and semi-minor axes, respectively.

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

A: To find the perimeter of an ellipse, you need to use the formula $P = 2\pi\sqrt{\frac{a^2+b^2}{2}}$ , where $a$ and $b$ are the lengths of the semi-major and semi-minor axes, respectively.

Q: Can an ellipse be a perfect circle?

A: Yes, an ellipse can be a perfect circle. In this case, the semi-major and semi-minor axes are equal, and the equation of the ellipse is $x^2 + y^2 = r^2$ , where $r$ is the radius of the circle.