The Vertices of the Ellipse

Oct 13, 2025 by miodeya.com 54 views

Introduction

An ellipse is a fundamental concept in mathematics, and it is essential to understand its properties and characteristics. In this article, we will focus on the equation of an ellipse and how to find its vertices. 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.

The Given Equation

The given equation is:

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

This equation represents an ellipse with its center at $(7, -2)$ . To find the vertices of the ellipse, we need to determine the values of $a$ and $b$ .

Finding the Values of $a$ and $b$

From the given equation, we can see that the denominator of the first term is $64$ , and the denominator of the second term is $9$ . Therefore, we can conclude that:

$a^2 = 64$ $b^2 = 9$

Taking the square root of both sides, we get:

$a = \sqrt{64} = 8$ $b = \sqrt{9} = 3$

The Vertices of the Ellipse

The vertices of the ellipse are located at a distance of $a$ units from the center along the major axis and at a distance of $b$ units from the center along the minor axis. Since the center of the ellipse is at $(7, -2)$ , the vertices will be located at:

$(7 + 8, -2) = (15, -2)$ $(7 - 8, -2) = (-1, -2)$

Therefore, the vertices of the ellipse are $(15, -2)$ and $(-1, -2)$ .

Conclusion

In this article, we have discussed the equation of an ellipse and how to find its vertices. We have used the given equation to determine the values of $a$ and $b$ and then located the vertices of the ellipse. The vertices of the ellipse are $(15, -2)$ and $(-1, -2)$ .

Answer

The correct answer is:

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

Additional Information

The equation of an ellipse can be written in the form $\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1$ , 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.
The vertices of an ellipse are located at a distance of $a$ units from the center along the major axis and at a distance of $b$ units from the center along the minor axis.
To find the vertices of an ellipse, we need to determine the values of $a$ and $b$ from the given equation.

References

[1] "Ellipses" by Math Open Reference. Retrieved from https://www.mathopenref.com/ellipse.html
[2] "Equation of an Ellipse" by Purplemath. Retrieved from https://www.purplemath.com/modules/ellipse.htm
The Equation of an Ellipse: Q&A =====================================

Introduction

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

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 model various real-world phenomena.

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

A: The standard form of the equation of an ellipse is:

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

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: How do I find the vertices of an ellipse?

A: To find the vertices of an ellipse, you need to determine the values of $a$ and $b$ from the given equation. The vertices are located at a distance of $a$ units from the center along the major axis and at a distance of $b$ units from the center along the minor axis.

Q: What is the difference between the major and minor axes of an ellipse?

A: The major axis of an ellipse is the longest diameter of the ellipse, while the minor axis is the shortest diameter. The major axis is always longer than the minor axis.

Q: How do I determine the center of an ellipse?

A: To determine the center of an ellipse, you need to look at the given equation and identify the values of $h$ and $k$ . The center of the ellipse is given by the point $(h,k)$ .

Q: What is the significance of the semi-major and semi-minor axes of an ellipse?

A: The semi-major and semi-minor axes of an ellipse are important because they determine the shape and size of the ellipse. The length of the semi-major axis is always greater than or equal to the length of the semi-minor axis.

Q: Can an ellipse have a semi-major axis that is equal to the semi-minor axis?

A: Yes, an ellipse can have a semi-major axis that is equal to the semi-minor axis. In this case, the ellipse is a circle.

Q: How do I graph an ellipse?

A: To graph an ellipse, you need to plot the center of the ellipse and then draw the major and minor axes. The vertices of the ellipse are located at a distance of $a$ units from the center along the major axis and at a distance of $b$ units from the center along the minor axis.

Q: What are some real-world applications of ellipses?

A: Ellipses have many real-world applications, including:

Modeling the orbits of planets and other celestial bodies
Describing the shape of a mirror or a lens
Modeling the shape of a satellite dish
Describing the shape of a baseball or a football

Conclusion

In this article, we have provided a Q&A section to help you better understand the concept of an ellipse and its properties. We hope that this article has been helpful in clarifying any doubts you may have had about ellipses.

Additional Information

Ellipses are used in many real-world applications, including physics, engineering, and computer science.
The equation of an ellipse can be written in the form $\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1$ , 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.
The vertices of an ellipse are located at a distance of $a$ units from the center along the major axis and at a distance of $b$ units from the center along the minor axis.

References

[1] "Ellipses" by Math Open Reference. Retrieved from https://www.mathopenref.com/ellipse.html
[2] "Equation of an Ellipse" by Purplemath. Retrieved from https://www.purplemath.com/modules/ellipse.htm