Find the equation of an ellipse given center, and the length of major and minor axes | @ProfD ​

Prof D
18 Apr 202104:26
EducationalLearning
32 Likes 10 Comments

TLDRIn this educational video, the presenter guides viewers through the process of finding the equation of an ellipse in standard form, given its center and the lengths of its axes. The video begins with the center coordinates (5, -2), a major axis of length 16 parallel to the y-axis, and a minor axis of length 8. The presenter explains that since the major axis is vertical, the ellipse is vertical as well. Using the lengths of the axes, the video derives the values for 'a' and 'b', which are 8 and 4, respectively. The final equation of the ellipse is presented as (x - 5)^2 / 16 + (y + 2)^2 / 64 = 1. This concise tutorial is ideal for those looking to understand the mathematical principles behind ellipses and their standard form equations.

Takeaways
  • 📚 The video is a tutorial on finding the equation of an ellipse in standard form.
  • 📐 The ellipse's center is given as (5, -2), located in the fourth quadrant.
  • 📏 The major axis of the ellipse is 16 units long and is parallel to the y-axis.
  • 📏 The minor axis is 8 units long, making it half the length of the major axis.
  • 🔍 The ellipse is vertical because the major axis is parallel to the y-axis.
  • 📝 The equation for a vertical ellipse is \((x-h)^2/(b^2) + (y-k)^2/(a^2) = 1\).
  • 🔢 The value of 'a' is calculated to be 8 by setting \(2a = 16\) and solving for 'a'.
  • 🔢 The value of 'b' is calculated to be 4 by setting \(2b = 8\) and solving for 'b'.
  • 📝 The final equation of the ellipse is \((x-5)^2/16 + (y+2)^2/64 = 1\).
  • 👨‍🏫 The video is presented by Prof D, who encourages questions in the comment section.
  • 👋 The video concludes with a sign-off from Prof D, indicating the end of the tutorial.
Q & A

  • What is the main topic of the video?

    -The main topic of the video is to demonstrate how to find the equation of an ellipse in standard form using its center and the lengths of its axes.

  • What are the coordinates of the center of the ellipse given in the video?

    -The center of the ellipse is given as (5, -2), which is in the fourth quadrant.

  • What is the length of the major axis of the ellipse mentioned in the video?

    -The length of the major axis of the ellipse is 16 units, which is parallel to the y-axis.

  • What is the length of the minor axis of the ellipse mentioned in the video?

    -The length of the minor axis of the ellipse is 8 units.

  • What type of ellipse is being discussed in the video?

    -The video discusses a vertical ellipse, which has its major axis parallel to the y-axis.

  • What is the general equation of a vertical ellipse in standard form?

    -The general equation of a vertical ellipse in standard form is \(\frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1\), where \(h\) and \(k\) are the coordinates of the center.

  • How is the value of 'a' determined from the major axis length in the video?

    -The value of 'a' is determined by the formula \(2a = \text{length of the major axis}\). In the video, \(2a = 16\), so \(a = 8\).

  • How is the value of 'b' determined from the minor axis length in the video?

    -The value of 'b' is determined by the formula \(2b = \text{length of the minor axis}\). In the video, \(2b = 8\), so \(b = 4\).

  • What is the final equation of the ellipse in standard form as given in the video?

    -The final equation of the ellipse in standard form is \(\frac{(x-5)^2}{4^2} + \frac{(y+2)^2}{8^2} = 1\), which simplifies to \(\frac{(x-5)^2}{16} + \frac{(y+2)^2}{64} = 1\).

  • Who is the presenter of the video?

    -The presenter of the video is Prof D.

  • What does the presenter suggest viewers do if they have questions or need clarifications?

    -The presenter suggests that viewers should put their questions or clarifications in the comment section below the video.

Outlines
00:00
📚 Introduction to Finding the Ellipse Equation

The video begins with a warm welcome and an introduction to the topic of finding the equation of an ellipse in standard form. The instructor outlines the prerequisites for the lesson, which include knowing the center of the ellipse and the lengths of its major and minor axes. The specific example provided has the ellipse centered at (5, -2) with a major axis of length 16 parallel to the y-axis and a minor axis of length 8.

📐 Understanding the Ellipse's Orientation and Axes

The instructor explains that the ellipse is vertical since its major axis is parallel to the y-axis, which means it's oriented vertically on the coordinate plane. The center of the ellipse is in the fourth quadrant, with coordinates (5, -2). The major axis is the longest diameter of the ellipse, and in this case, it is given as 16 units long, which is twice the length of the semi-major axis (2a = 16). The minor axis is the shortest diameter and is given as 8 units long, which is twice the length of the semi-minor axis (2b = 8).

🔍 Calculating the Semi-Major and Semi-Minor Axes

The video proceeds with the calculation of the semi-major axis (a) and the semi-minor axis (b) of the ellipse. The instructor divides the length of the major axis by 2 to find that a = 8. Similarly, the length of the minor axis is divided by 2 to find that b = 4. These calculations are crucial for determining the coefficients in the standard form equation of the ellipse.

📝 Deriving the Standard Form Equation of the Ellipse

With the values of a and b determined, the instructor derives the standard form equation of the ellipse. For a vertical ellipse, the equation is given by (x - h)²/b² + (y - k)²/a² = 1, where (h, k) is the center of the ellipse. Substituting the values of h = 5, k = -2, a = 8, and b = 4 into the equation, the instructor simplifies it to (x - 5)²/16 + (y + 2)²/64 = 1, which represents the ellipse in standard form.

👋 Conclusion and Invitation for Feedback

The video concludes with a summary of the process and an invitation for viewers to ask questions or seek clarifications in the comment section below. The instructor, Prof D, thanks the viewers for watching and signs off with a friendly farewell, indicating the end of the educational content.

Mindmap
Keywords
💡Ellipse
An ellipse is a geometric shape that resembles a flattened circle. In the script, the ellipse is the main focus of the video, where the presenter is teaching the viewers how to find its equation. The ellipse is characterized by its two axes, the major and minor axes, which are used to derive its standard form equation.
💡Standard Form
The standard form of an ellipse's equation is a specific way to represent the ellipse mathematically. It is defined as (x-h)^2/a^2 + (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 the script, the presenter derives the standard form equation for the given ellipse.
💡Center
The center of an ellipse is the point equidistant from all points on the ellipse, and it is represented by the coordinates (h, k). In the provided script, the center of the ellipse is given as (5, -2), which is used to set up the standard form equation of the ellipse.
💡Major Axis
The major axis of an ellipse is the longest diameter of the ellipse, passing through its center. It is parallel to the longer of the two axes. In the video script, the major axis has a length of 16 and is parallel to the y-axis, indicating a vertical ellipse.
💡Minor Axis
The minor axis is the shortest diameter of an ellipse, perpendicular to the major axis. It is used to determine the value of b in the standard form equation. In the script, the minor axis has a length of 8, which is half of its total length, making 2b equal to 8.
💡Axis Length
The length of an axis is a measure of the distance between the two points where the axis intersects the ellipse. In the context of the video, the lengths of the major and minor axes are given as 16 and 8, respectively, and are crucial for finding the equation of the ellipse.
💡Vertical Ellipse
A vertical ellipse is an ellipse where the major axis is oriented vertically (parallel to the y-axis). The script describes the ellipse in question as a vertical ellipse, which affects the orientation of its standard form equation.
💡Equation Derivation
Equation derivation is the process of finding the mathematical equation that represents a geometric shape. In the video, the presenter goes through the steps of deriving the equation of the ellipse using its center, major axis length, and minor axis length.
💡Semi-major Axis (a)
The semi-major axis (a) is half the length of the major axis of an ellipse. It is used in the standard form equation to determine the shape's elongation. In the script, the length of the major axis is 16, so the semi-major axis (a) is calculated to be 8.
💡Semi-minor Axis (b)
The semi-minor axis (b) is half the length of the minor axis of an ellipse. It is also used in the standard form equation and, together with the semi-major axis, defines the ellipse's dimensions. From the script, the semi-minor axis (b) is found to be 4.
💡Prof D
Prof D appears to be the presenter or the host of the video channel. They guide the viewers through the process of finding the equation of an ellipse, providing explanations and mathematical steps. The mention of 'Prof D' at the end of the script suggests that they are signing off and closing the video.
Highlights

Introduction to finding the equation of an ellipse in standard form.

Providing the center of the ellipse at coordinates (5, -2).

Identifying the major axis length as 16 and parallel to the y-axis.

Determining the minor axis length as 8.

Explanation of the ellipse being in the fourth quadrant.

Clarification that the ellipse is vertical due to the major axis orientation.

Introduction of the standard equation for a vertical ellipse.

Substitution of the center coordinates into the ellipse equation.

Calculation of the semi-major axis length (a) as 8.

Calculation of the semi-minor axis length (b) as 4.

Substitution of axis lengths into the ellipse equation.

Final simplification of the ellipse equation in standard form.

Presentation of the complete equation: (x-5)^2/16 + (y+2)^2/64 = 1.

Invitation for viewers to ask questions or seek clarifications.

Closing remarks and sign-off by Prof D.

Transcripts

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Ellipse EquationMathematics TutorialStandard FormCenter CoordinatesAxis LengthsVertical EllipseEducational ContentGeometry LessonsMath ChannelEquation DerivationOnline LearningMathematical Concepts