Identifying the Key Components of the Ellipse

Oct 13, 2025 by windailys.com 38 views

An ellipse is a fundamental concept in mathematics, particularly 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 standard form of an ellipse centered at the origin is given by the equation ${\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1}$ , where ${a}$ and ${b}$ are the lengths of the semi-major and semi-minor axes, respectively.

Identifying the Key Components of the Ellipse

In this problem, we are given the vertices of the ellipse as ${(-5,0)}$ and ${(5,0)}$ , which indicates that the major axis of the ellipse lies along the x-axis. The center of the ellipse is at the origin, and the point ${(0,2)}$ lies on the ellipse. We need to determine the equation of the ellipse using this information.

Determining the Lengths of the Semi-Major and Semi-Minor Axes

The distance between the vertices of the ellipse is equal to the length of the major axis, which is ${2a}$ . In this case, the distance between ${(-5,0)}$ and ${(5,0)}$ is ${10}$ , so we have ${2a = 10}$ . Solving for ${a}$ , we get ${a = 5}$ .

Finding the Equation of the Ellipse

Now that we have the value of ${a}$ , we can use the point ${(0,2)}$ to find the value of ${b}$ . Substituting the values of ${x}$ and ${y}$ into the equation of the ellipse, we get ${\frac{0^2}{5^2} + \frac{2^2}{b^2} = 1}$ . Simplifying the equation, we get ${\frac{4}{b^2} = 1}$ . Solving for ${b^2}$ , we get ${b^2 = 4}$ . Taking the square root of both sides, we get ${b = 2}$ .

Writing the Equation of the Ellipse

Now that we have the values of ${a}$ and ${b}$ , we can write the equation of the ellipse. The standard form of the equation of an ellipse centered at the origin is ${\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1}$ . Substituting the values of ${a}$ and ${b}$ , we get ${\frac{x^2}{5^2} + \frac{y^2}{2^2} = 1}$ . Simplifying the equation, we get ${\frac{x^2}{25} + \frac{y^2}{4} = 1}$ .

Conclusion

In this problem, we were given the vertices of an ellipse and a point on the ellipse. We used this information to determine the equation of the ellipse. We found the lengths of the semi-major and semi-minor axes and used the point on the ellipse to find the value of the semi-minor axis. Finally, we wrote the equation of the ellipse in standard form.

Key Takeaways

The standard form of the equation of an ellipse centered at the origin is ${\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1}$ .
The length of the semi-major axis is equal to the distance between the vertices of the ellipse.
The length of the semi-minor axis can be found using the point on the ellipse.

Common Mistakes to Avoid

Failing to identify the key components of the ellipse, such as the center and the vertices.
Not using the point on the ellipse to find the value of the semi-minor axis.
Not writing the equation of the ellipse in standard form.

Real-World Applications

Ellipses are used in various real-world applications, such as astronomy, engineering, and physics.
They are used to model the orbits of planets and other celestial bodies.
They are used to design optical systems, such as telescopes and microscopes.

Practice Problems

Find the equation of an ellipse with vertices ${(-3,0)}$ and ${(3,0)}$ and a point on the ellipse ${(0,4)}$ .
Find the equation of an ellipse with center ${(2,0)}$ and vertices ${(-1,0)}$ and ${(5,0)}$ and a point on the ellipse ${(3,2)}$ .

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 centered at the origin is ${\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1}$ , where ${a}$ and ${b}$ are the lengths of the semi-major and semi-minor axes, respectively.

Q: How do I find the lengths of the semi-major and semi-minor axes?

A: To find the lengths of the semi-major and semi-minor axes, you need to identify the key components of the ellipse, such as the center and the vertices. The length of the semi-major axis is equal to the distance between the vertices of the ellipse, and the length of the semi-minor axis can be found using the point on the ellipse.

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

A: To write the equation of an ellipse, you need to use the standard form of the equation of an ellipse and substitute the values of the semi-major and semi-minor axes. You also need to use the point on the ellipse to find the value of the semi-minor axis.

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

A: Some common mistakes to avoid when working with ellipses include failing to identify the key components of the ellipse, not using the point on the ellipse to find the value of the semi-minor axis, and not writing the equation of the ellipse in standard form.

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

A: Ellipses are used in various real-world applications, such as astronomy, engineering, and physics. They are used to model the orbits of planets and other celestial bodies, and they are used to design optical systems, such as telescopes and microscopes.

Q: How do I find the equation of an ellipse with a given center and vertices?

A: To find the equation of an ellipse with a given center and vertices, you need to use the standard form of the equation of an ellipse and substitute the values of the center and the vertices. You also need to use the point on the ellipse to find the value of the semi-minor axis.

Q: How do I find the equation of an ellipse with a given center and a point on the ellipse?

A: To find the equation of an ellipse with a given center and a point on the ellipse, you need to use the standard form of the equation of an ellipse and substitute the values of the center and the point on the ellipse. You also need to use the point on the ellipse to find the value of the semi-minor axis.

Q: What are some practice problems for working with ellipses?

A: Some practice problems for working with ellipses include finding the equation of an ellipse with vertices ${(-3,0)}$ and ${(3,0)}$ and a point on the ellipse ${(0,4)}$ , and finding the equation of an ellipse with center ${(2,0)}$ and vertices ${(-1,0)}$ and ${(5,0)}$ and a point on the ellipse ${(3,2)}$ .

Q: How do I check my work when working with ellipses?

A: To check your work when working with ellipses, you need to verify that the equation of the ellipse satisfies the standard form of the equation of an ellipse and that the lengths of the semi-major and semi-minor axes are correct.

Q: What are some common errors to look out for when working with ellipses?

A: Some common errors to look out for when working with ellipses include failing to identify the key components of the ellipse, not using the point on the ellipse to find the value of the semi-minor axis, and not writing the equation of the ellipse in standard form.

Q: How do I use technology to work with ellipses?

A: You can use technology, such as graphing calculators or computer software, to work with ellipses. These tools can help you visualize the ellipse and check your work.

Q: What are some advanced topics in ellipses?

A: Some advanced topics in ellipses include the equation of an ellipse in a rotated coordinate system, the equation of an ellipse with a given center and a point on the ellipse, and the equation of an ellipse with a given center and vertices.