An Ellipse Has A Center At The Origin, A Vertex Along The Major Axis At \[$(10,0)\$\], And A Focus At \[$(8,0)\$\].Which Equation Represents This Ellipse?A. \[$\frac{x^2}{10^2} + \frac{y^2}{6^2} = 1\$\]B. \[$\frac{x^2}{10^2}

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. In this article, we will explore how to find the equation of an ellipse with a center at the origin, a vertex along the major axis at (10,0), and a focus at (8,0).

Understanding the Properties of an Ellipse

To find the equation of the ellipse, we need to understand its properties. The center of the ellipse is at the origin (0,0), which means that the ellipse is symmetric about the x-axis. The vertex along the major axis is at (10,0), which indicates that the major axis is along the x-axis. The focus is at (8,0), which is 2 units away from the center.

The General Equation of an Ellipse

The general equation of an ellipse with its center at the origin is given by:

$$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$

where $a$ is the length of the semi-major axis and $b$ is the length of the semi-minor axis.

Finding the Length of the Semi-Major Axis

The length of the semi-major axis is the distance from the center to the vertex along the major axis. In this case, the vertex is at (10,0), so the length of the semi-major axis is 10.

Finding the Length of the Semi-Minor Axis

To find the length of the semi-minor axis, we need to use the relationship between the semi-major axis, semi-minor axis, and the distance from the center to the focus. This relationship is given by:

$$c^2 = a^2 - b^2$$

where $c$ is the distance from the center to the focus.

In this case, the distance from the center to the focus is 2, so we have:

$$2^2 = 10^2 - b^2$$

Simplifying, we get:

$$4 = 100 - b^2$$

Solving for $b^2$, we get:

$$b^2 = 96$$

Taking the square root, we get:

$$b = \sqrt{96} = 4\sqrt{6}$$

Finding the Equation of the Ellipse

Now that we have found the lengths of the semi-major axis and semi-minor axis, we can find the equation of the ellipse. Substituting the values of $a^2$ and $b^2$ into the general equation of the ellipse, we get:

$$\frac{x^2}{10^2} + \frac{y^2}{(4\sqrt{6})^2} = 1$$

Simplifying, we get:

$$\frac{x^2}{100} + \frac{y^2}{96} = 1$$

Conclusion

In this article, we have found the equation of an ellipse with a center at the origin, a vertex along the major axis at (10,0), and a focus at (8,0). We have used the properties of the ellipse to find the lengths of the semi-major axis and semi-minor axis, and then used these values to find the equation of the ellipse.

The Final Answer

The equation of the ellipse is:

$$\frac{x^2}{100} + \frac{y^2}{96} = 1$$

This equation represents the ellipse with the given properties.

Comparison with the Options

Let's compare the equation we found with the options given:

A. $\frac{x^2}{10^2} + \frac{y^2}{6^2} = 1$

B. $\frac{x^2}{10^2} + \frac{y^2}{(4\sqrt{6})^2} = 1$

C. $\frac{x^2}{100} + \frac{y^2}{96} = 1$

D. $\frac{x^2}{100} + \frac{y^2}{(4\sqrt{6})^2} = 1$

The correct equation is option C.

Final Thoughts

Q: What is the general equation of an ellipse with its center at the origin?

A: The general equation of an ellipse with its center at the origin is given by:

$$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$

where $a$ is the length of the semi-major axis and $b$ is the length of the semi-minor axis.

Q: How do I find the length of the semi-major axis?

A: To find the length of the semi-major axis, you need to find the distance from the center to the vertex along the major axis. In the case of the ellipse with a center at the origin, a vertex along the major axis at (10,0), and a focus at (8,0), the length of the semi-major axis is 10.

Q: How do I find the length of the semi-minor axis?

A: To find the length of the semi-minor axis, you need to use the relationship between the semi-major axis, semi-minor axis, and the distance from the center to the focus. This relationship is given by:

$$c^2 = a^2 - b^2$$

where $c$ is the distance from the center to the focus.

Q: What is the relationship between the semi-major axis, semi-minor axis, and the distance from the center to the focus?

A: The relationship between the semi-major axis, semi-minor axis, and the distance from the center to the focus is given by:

$$c^2 = a^2 - b^2$$

where $c$ is the distance from the center to the focus.

Q: How do I find the equation of the ellipse?

A: To find the equation of the ellipse, you need to substitute the values of $a^2$ and $b^2$ into the general equation of the ellipse. In the case of the ellipse with a center at the origin, a vertex along the major axis at (10,0), and a focus at (8,0), the equation of the ellipse is:

$$\frac{x^2}{100} + \frac{y^2}{96} = 1$$

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

A: The semi-major axis and semi-minor axis are the lengths of the two axes of the ellipse. The semi-major axis is the length of the longest diameter of the ellipse, while the semi-minor axis is the length of the shortest diameter of the ellipse.

Q: How do I determine the orientation of the major axis of an ellipse?

A: To determine the orientation of the major axis of an ellipse, you need to find the vertex along the major axis. In the case of the ellipse with a center at the origin, a vertex along the major axis at (10,0), and a focus at (8,0), the major axis is along the x-axis.

Q: What is the significance of 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 used to define the shape of the ellipse and are used in the equation of the ellipse.

Q: How do I find the distance from the center to the focus of an ellipse?

A: To find the distance from the center to the focus of an ellipse, you need to use the relationship between the semi-major axis, semi-minor axis, and the distance from the center to the focus. This relationship is given by:

$$c^2 = a^2 - b^2$$

where $c$ is the distance from the center to the focus.

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

A: The eccentricity of an ellipse is a measure of how elliptical the ellipse is. The eccentricity is defined as the ratio of the distance from the center to the focus to 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 relationship between the semi-major axis, semi-minor axis, and the distance from the center to the focus. This relationship is given by:

$$c^2 = a^2 - b^2$$

where $c$ is the distance from the center to the focus.

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

A: The equation of an ellipse is a mathematical representation of the shape of the ellipse. The equation is used to define the shape of the ellipse and is used in various applications such as physics, engineering, and computer science.

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

A: The equation of an ellipse is used in various real-world applications such as physics, engineering, and computer science. For example, the equation of an ellipse is used to model the shape of a satellite orbit, the shape of a mirror, and the shape of a lens.

Q: What are some common applications of the equation of an ellipse?

A: Some common applications of the equation of an ellipse include:

• Modeling the shape of a satellite orbit
• Modeling the shape of a mirror
• Modeling the shape of a lens
• Modeling the shape of a telescope
• Modeling the shape of a camera

Q: How do I graph an ellipse using the equation of an ellipse?

A: To graph an ellipse using the equation of an ellipse, you need to substitute the values of $a^2$ and $b^2$ into the general equation of the ellipse. Then, you need to plot the points on a coordinate plane and connect the points to form the ellipse.

Q: What are some common mistakes to avoid when graphing an ellipse?

A: Some common mistakes to avoid when graphing an ellipse include:

• Not using the correct values of $a^2$ and $b^2$
• Not plotting the points on a coordinate plane
• Not connecting the points to form the ellipse
• Not using the correct scale on the coordinate plane

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 $A$ is the area of the ellipse, $a$ is the length of the semi-major axis, and $b$ is the length of the semi-minor axis.

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

A: The area of an ellipse is a measure of the size of the ellipse. The area is used in various applications such as physics, engineering, and computer science.

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 $P$ is the perimeter of the ellipse, $a$ is the length of the semi-major axis, and $b$ is the length of the semi-minor axis.

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

A: The perimeter of an ellipse is a measure of the size of the ellipse. The perimeter is used in various applications such as physics, engineering, and computer science.