The Equation ( X + 1 ) 2 225 + ( Y + 6 ) 2 144 = 1 \frac{(x+1)^2}{225}+\frac{(y+6)^2}{144}=1 225 ( X + 1 ) 2 + 144 ( Y + 6 ) 2 = 1 Represents An Ellipse.Which Points Are The Foci Of The Ellipse?

Mar 11, 2025 by ADMIN 199 views

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

Introduction

An ellipse is a fundamental concept in mathematics, and it has numerous applications in various fields, including physics, engineering, and computer science. The equation of an ellipse 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 semi-major and semi-minor axes, respectively. In this article, we will focus on finding the foci of an ellipse represented by the equation $\frac{(x+1)^2}{225}+\frac{(y+6)^2}{144}=1$ .

Understanding the Equation

The given equation is in the standard form of an ellipse, where the center is $(-1,-6)$ , and the semi-major and semi-minor axes are $\sqrt{225}=15$ and $\sqrt{144}=12$ , respectively. To find the foci of the ellipse, we need to calculate the distance between the center and each focus, which is given by $c=\sqrt{a^2-b^2}$ .

Calculating the Distance

Using the values of $a$ and $b$ , we can calculate the distance $c$ as follows:

$c=\sqrt{a^2-b^2}=\sqrt{15^2-12^2}=\sqrt{225-144}=\sqrt{81}=9$

Finding the Foci

The foci of the ellipse are located at a distance $c$ from the center, along the major axis. Since the major axis is horizontal, the foci will be located at $(h\pm c,k)$ . Substituting the values of $h$ , $k$ , and $c$ , we get:

$(h\pm c,k)=(-1\pm 9,-6)$

Therefore, the foci of the ellipse are located at $(-1+9,-6)=(8,-6)$ and $(-1-9,-6)=(-10,-6)$ .

Conclusion

In this article, we have found the foci of an ellipse represented by the equation $\frac{(x+1)^2}{225}+\frac{(y+6)^2}{144}=1$ . We have calculated the distance between the center and each focus using the formula $c=\sqrt{a^2-b^2}$ and have determined the coordinates of the foci. The foci of the ellipse are located at $(8,-6)$ and $(-10,-6)$ .

Key Takeaways

The equation of an ellipse is given by $\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1$ .
The center of the ellipse is $(h,k)$ , and the semi-major and semi-minor axes are $a$ and $b$ , respectively.
The distance between the center and each focus is given by $c=\sqrt{a^2-b^2}$ .
The foci of the ellipse are located at $(h\pm c,k)$ .

References

[1] "Geometry: A Comprehensive Introduction" by Dan Pedoe
[2] "Mathematics for Computer Science" by Eric Lehman, F Thomson Leighton, and Albert R Meyer

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 has numerous applications in various fields.

Q: What is the equation of an ellipse?

A: The equation of an ellipse 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 semi-major and semi-minor axes, respectively.

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

A: To find the foci of an ellipse, you need to calculate the distance between the center and each focus, which is given by $c=\sqrt{a^2-b^2}$ . The foci are located at $(h\pm c,k)$ .

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

A: The foci of an ellipse are significant because they determine the shape and size of the ellipse. The distance between the foci and the center of the ellipse is related to the eccentricity of the ellipse, which is a measure of how elliptical the shape is.

Q: Can I find the foci of an ellipse using a calculator?

A: Yes, you can find the foci of an ellipse using a calculator. Simply enter the values of $a$ and $b$ into the calculator and use the formula $c=\sqrt{a^2-b^2}$ to calculate the distance between the center and each focus.

Q: How do I graph an ellipse?

A: To graph an ellipse, you need to plot the center of the ellipse and the foci. Then, draw a curve that passes through the center and the foci, and is symmetrical about the center.

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

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

Astronomy: Ellipses are used to model the orbits of planets and other celestial bodies.
Physics: Ellipses are used to describe the motion of objects under the influence of a central force.
Engineering: Ellipses are used to design curves and shapes for various applications, such as bridges and buildings.
Computer Science: Ellipses are used in computer graphics and game development to create realistic shapes and movements.

Q: Can I use ellipses to model real-world phenomena?

A: Yes, you can use ellipses to model real-world phenomena, such as the motion of a pendulum, the orbit of a satellite, or the shape of a leaf.

Q: What are some common mistakes to avoid when working with ellipses?

A: Some common mistakes to avoid when working with ellipses include:

Confusing the semi-major and semi-minor axes: Make sure to identify the correct axes and use the correct values in your calculations.
Failing to calculate the distance between the center and each focus: Make sure to calculate the distance using the formula $c=\sqrt{a^2-b^2}$ .
Graphing the ellipse incorrectly: Make sure to plot the center and foci correctly and draw a curve that passes through the center and foci.

Q: Where can I learn more about ellipses and their applications?

A: You can learn more about ellipses and their applications by:

Reading books and online resources: There are many books and online resources available that provide detailed information on ellipses and their applications.
Taking online courses: Online courses can provide a comprehensive introduction to ellipses and their applications.
Joining online communities: Joining online communities, such as forums and social media groups, can provide a platform to ask questions and learn from others.

Q: What are some advanced topics related to ellipses?

A: Some advanced topics related to ellipses include:

Eccentricity: The eccentricity of an ellipse is a measure of how elliptical the shape is.
Conic sections: Conic sections are curves that result from the intersection of a cone and a plane.
Parametric equations: Parametric equations are used to describe the motion of objects in terms of parameters.

Note: The questions and answers provided are for illustrative purposes only and may not be directly related to the topic of finding the foci of an ellipse.