Given The Ellipse ( X − 3 ) 2 4 + ( Y − 4 ) 2 36 = 1 \frac{(x-3)^2}{4}+\frac{(y-4)^2}{36}=1 4 ( X − 3 ) 2 ​ + 36 ( Y − 4 ) 2 ​ = 1 ,1. Find The Center Point: $\square$2. List The Vertices (separated By A Comma): □ \square □

An ellipse is a fundamental concept in mathematics, and it is essential to understand its properties and characteristics. In this article, we will delve into the world of ellipses and explore how to find the center point and vertices of a given ellipse.

What is an Ellipse?

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 type of conic section, which is a curve obtained by intersecting a cone with a plane. Ellipses have various applications in mathematics, physics, engineering, and other fields.

The Standard Form of an Ellipse

The standard form of an ellipse is given by the equation:

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

where $(h,k)$ is the center point of the ellipse, and $a$ and $b$ are the lengths of the semi-major and semi-minor axes, respectively.

Finding the Center Point of the Ellipse

To find the center point of the ellipse, we need to look at the equation of the ellipse and identify the values of $h$ and $k$. In the given equation:

$$\frac{(x-3)^2}{4} + \frac{(y-4)^2}{36} = 1$$

we can see that $h=3$ and $k=4$. Therefore, the center point of the ellipse is $(3,4)$.

Listing the Vertices of the Ellipse

To list the vertices of the ellipse, we need to find the points where the ellipse intersects the major and minor axes. The vertices of the ellipse are located at a distance of $a$ and $b$ from the center point, respectively.

In the given equation, $a^2=4$ and $b^2=36$. Therefore, $a=2$ and $b=6$. The vertices of the ellipse are located at $(h\pm a, k)$ and $(h, k\pm b)$.

The vertices of the ellipse are:

$(3+2, 4) = (5, 4)$ $(3-2, 4) = (1, 4)$ $(3, 4+6) = (3, 10)$ $(3, 4-6) = (3, -2)$

Therefore, the vertices of the ellipse are $(5, 4), (1, 4), (3, 10),$ and $(3, -2)$.

Conclusion

In this article, we have explored the concept of ellipses and how to find the center point and vertices of a given ellipse. We have used the standard form of an ellipse and identified the values of $h$, $k$, $a$, and $b$ to find the center point and vertices of the ellipse. We have also listed the vertices of the ellipse and provided a clear understanding of the properties and characteristics of ellipses.

Frequently Asked Questions

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 an ellipse?

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

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

Q: How do I find the center point of an ellipse?

A: To find the center point of an ellipse, you need to look at the equation of the ellipse and identify the values of $h$ and $k$.

Q: How do I list the vertices of an ellipse?

A: To list the vertices of an ellipse, you need to find the points where the ellipse intersects the major and minor axes.

Q: What are the vertices of the given ellipse?

A: The vertices of the given ellipse are $(5, 4), (1, 4), (3, 10),$ and $(3, -2)$.

References

• [1] "Ellipses" by Math Open Reference. Retrieved from https://www.mathopenref.com/ellipse.html
• [2] "Conic Sections" by Khan Academy. Retrieved from https://www.khanacademy.org/math/geometry/conic-sections

Glossary

• Ellipse: A closed curve on a plane surrounding two focal points such that the sum of the distances to the two focal points is constant.
• Center point: The point at the center of the ellipse.
• Vertices: The points where the ellipse intersects the major and minor axes.
• Semi-major axis: The length of the semi-major axis of the ellipse.
• Semi-minor axis: The length of the semi-minor axis of the ellipse.
  Ellipses: A Comprehensive Q&A Guide =====================================

In our previous article, we explored the concept of ellipses and how to find the center point and vertices of a given ellipse. In this article, we will provide a comprehensive Q&A guide to help you understand ellipses better.

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 an ellipse?

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

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

where $(h,k)$ is the center point 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 center point of an ellipse?

A: To find the center point of an ellipse, you need to look at the equation of the ellipse and identify the values of $h$ and $k$.

Q: How do I list the vertices of an ellipse?

A: To list the vertices of an ellipse, you need to find the points where the ellipse intersects the major and minor axes.

Q: What are the vertices of the given ellipse?

A: The vertices of the given ellipse are $(5, 4), (1, 4), (3, 10),$ and $(3, -2)$.

Q: What is the difference between a circle and an ellipse?

A: A circle is a special type of ellipse where the semi-major and semi-minor axes are equal. In other words, a circle is an ellipse with $a=b$.

Q: Can an ellipse have a negative value for $a$ or $b$?

A: No, an ellipse cannot have a negative value for $a$ or $b$. The values of $a$ and $b$ must be positive.

Q: How do I graph an ellipse?

A: To graph an ellipse, you need to plot the center point and the vertices of the ellipse. Then, draw a smooth curve through the center point and the vertices.

Q: What is the equation of an ellipse in the coordinate plane?

A: The equation of an ellipse in the coordinate plane is given by:

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

Q: Can an ellipse be rotated in the coordinate plane?

A: Yes, an ellipse can be rotated in the coordinate plane. The equation of a rotated ellipse is given by:

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

where $\theta$ is the angle of rotation.

Q: What is the foci of an ellipse?

A: The foci of an ellipse are the two points inside the ellipse that are equidistant from the center point.

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

A: To find the foci of an ellipse, you need to use the equation:

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

where $c$ is the distance from the center point to the foci.

Q: What is the eccentricity of an ellipse?

A: The eccentricity of an ellipse is a measure of how elliptical the ellipse is. It is given by:

$$e = \frac{c}{a}$$

where $c$ is the distance from the center point to the foci, and $a$ is the length of the semi-major axis.

Q: Can an ellipse have a negative eccentricity?

A: No, an ellipse cannot have a negative eccentricity. The eccentricity of an ellipse must be between 0 and 1.

Conclusion

In this article, we have provided a comprehensive Q&A guide to help you understand ellipses better. We have covered topics such as the standard form of an ellipse, finding the center point and vertices of an ellipse, graphing an ellipse, and more.

Frequently Asked Questions

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 an ellipse?

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

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

Q: How do I find the center point of an ellipse?

A: To find the center point of an ellipse, you need to look at the equation of the ellipse and identify the values of $h$ and $k$.

Q: How do I list the vertices of an ellipse?

A: To list the vertices of an ellipse, you need to find the points where the ellipse intersects the major and minor axes.

References

• [1] "Ellipses" by Math Open Reference. Retrieved from https://www.mathopenref.com/ellipse.html
• [2] "Conic Sections" by Khan Academy. Retrieved from https://www.khanacademy.org/math/geometry/conic-sections

Glossary

• Ellipse: A closed curve on a plane surrounding two focal points such that the sum of the distances to the two focal points is constant.
• Center point: The point at the center of the ellipse.
• Vertices: The points where the ellipse intersects the major and minor axes.
• Semi-major axis: The length of the semi-major axis of the ellipse.
• Semi-minor axis: The length of the semi-minor axis of the ellipse.
• Foci: The two points inside the ellipse that are equidistant from the center point.
• Eccentricity: A measure of how elliptical the ellipse is.