The Equation Of The Ellipse That Has A Center At { (5,4)$}$, A Focus At { (8,4)$}$, And A Vertex At { (10,4)$}$ Is:$ \frac{(x-C) 2}{A 2} + \frac{(y-D) 2}{B 2} = 1 }$where { A =$ $

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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 is a mathematical representation of this concept, and it is essential to understand how to derive and use it in various mathematical and real-world applications. In this article, we will focus on finding the equation of an ellipse with a given center, focus, and vertex.

Understanding the Components of an Ellipse

To derive the equation of an ellipse, we need to understand its components. The center of the ellipse is the point around which the ellipse is symmetric. The foci are two points inside the ellipse that help define its shape. The vertices are the points on the ellipse that are farthest from the center. In this case, we are given the center at (5,4), a focus at (8,4), and a vertex at (10,4).

Deriving the Equation of an Ellipse

The standard form of the equation of an ellipse is:

(x−h)2a2+(y−k)2b2=1\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.

To derive the equation of the ellipse, we need to find the values of a, b, and the center (h,k). We are given the center at (5,4), so h = 5 and k = 4.

Finding the Value of a

The distance between the center and the vertex is equal to the length of the semi-major axis (a). Since the vertex is at (10,4), we can find the value of a as follows:

a=(10−5)2+(4−4)2=25+0=25=5a = \sqrt{(10-5)^2 + (4-4)^2} = \sqrt{25 + 0} = \sqrt{25} = 5

Finding the Value of c

The distance between the center and the focus is equal to the value of c, which is given by:

c=(8−5)2+(4−4)2=9+0=9=3c = \sqrt{(8-5)^2 + (4-4)^2} = \sqrt{9 + 0} = \sqrt{9} = 3

Finding the Value of b

We can find the value of b using the relationship between a, b, and c:

b=a2−c2=52−32=25−9=16=4b = \sqrt{a^2 - c^2} = \sqrt{5^2 - 3^2} = \sqrt{25 - 9} = \sqrt{16} = 4

Writing the Equation of the Ellipse

Now that we have the values of a, b, and the center (h,k), we can write the equation of the ellipse as follows:

(x−5)252+(y−4)242=1\frac{(x-5)^2}{5^2} + \frac{(y-4)^2}{4^2} = 1

Simplifying the equation, we get:

(x−5)225+(y−4)216=1\frac{(x-5)^2}{25} + \frac{(y-4)^2}{16} = 1

Conclusion

In this article, we derived the equation of an ellipse with a given center, focus, and vertex. We used the standard form of the equation of an ellipse and found the values of a, b, and the center (h,k) using the given information. The equation of the ellipse is:

(x−5)225+(y−4)216=1\frac{(x-5)^2}{25} + \frac{(y-4)^2}{16} = 1

This equation represents the ellipse with a center at (5,4), a focus at (8,4), and a vertex at (10,4). We hope this article has provided a comprehensive guide to understanding the equation of an ellipse and its components.

Applications of the Equation of an Ellipse

The equation of an ellipse has numerous applications in mathematics, physics, and engineering. Some of the applications include:

  • Orbital Mechanics: The equation of an ellipse is used to describe the orbits of planets and other celestial bodies.
  • Optics: The equation of an ellipse is used to describe the shape of lenses and mirrors.
  • Electrical Engineering: The equation of an ellipse is used to describe the shape of electrical circuits.
  • Computer Graphics: The equation of an ellipse is used to create realistic images and animations.

Real-World Examples of Ellipses

Ellipses are found in various real-world applications, including:

  • Eggs: The shape of an egg is an example of an ellipse.
  • Teardrops: The shape of a teardrop is an example of an ellipse.
  • Mirrors: The shape of a mirror is an example of an ellipse.
  • Lenses: The shape of a lens is an example of an ellipse.

Conclusion

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.

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

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

(x−h)2a2+(y−k)2b2=1\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 values of a, b, and the center (h,k) for an ellipse?

A: To find the values of a, b, and the center (h,k), you need to know the coordinates of the center, a focus, and a vertex of the ellipse. You can then use the formulas:

a=(xvertex−h)2+(yvertex−k)2a = \sqrt{(x_{vertex} - h)^2 + (y_{vertex} - k)^2}

b=a2−c2b = \sqrt{a^2 - c^2}

c=(xfocus−h)2+(yfocus−k)2c = \sqrt{(x_{focus} - h)^2 + (y_{focus} - k)^2}

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

A: The semi-major axis (a) is the length of the longest diameter of the ellipse, while the semi-minor axis (b) is the length of the shortest diameter of the ellipse.

Q: How do I write the equation of an ellipse in standard form?

A: To write the equation of an ellipse in standard form, you need to know the values of a, b, and the center (h,k). You can then plug these values into the standard form of the equation:

(x−h)2a2+(y−k)2b2=1\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1

Q: What are some real-world applications of the equation of an ellipse?

A: The equation of an ellipse has numerous real-world applications, including:

  • Orbital Mechanics: The equation of an ellipse is used to describe the orbits of planets and other celestial bodies.
  • Optics: The equation of an ellipse is used to describe the shape of lenses and mirrors.
  • Electrical Engineering: The equation of an ellipse is used to describe the shape of electrical circuits.
  • Computer Graphics: The equation of an ellipse is used to create realistic images and animations.

Q: How do I graph an ellipse?

A: To graph an ellipse, you need to know the values of a, b, and the center (h,k). You can then use the following steps:

  1. Plot the center of the ellipse (h,k).
  2. Plot the vertices of the ellipse (h ± a, k).
  3. Plot the foci of the ellipse (h ± c, k).
  4. Draw the ellipse by connecting the vertices and foci.

Q: What are some common mistakes to avoid when working with the equation of an ellipse?

A: Some common mistakes to avoid when working with the equation of an ellipse include:

  • Not using the correct values of a, b, and the center (h,k).
  • Not using the correct formula for the equation of an ellipse.
  • Not graphing the ellipse correctly.

Q: How do I use the equation of an ellipse in real-world applications?

A: To use the equation of an ellipse in real-world applications, you need to understand the concept of an ellipse and how it is used in various fields. You can then apply the equation of an ellipse to solve problems and create models.

Conclusion

In conclusion, the equation of an ellipse is a fundamental concept in mathematics and has numerous real-world applications. We hope this FAQ article has provided a comprehensive guide to understanding the equation of an ellipse and its components.