How to find the equation of an ellipse given foci and vertices | @ProfD

Prof D
13 Apr 202105:30
EducationalLearning
32 Likes 10 Comments

TLDRIn this educational video, Prof D demonstrates how to derive the equation of an ellipse given its vertices and foci. The example provided has vertices at (±8, 0) and foci at (±5, 0), indicating a horizontally oriented major axis. The process involves calculating the semi-major axis (a) and semi-minor axis (b) using the distances of the vertices and foci. The formula \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) is then applied to find the ellipse's equation, resulting in \( \frac{x^2}{64} + \frac{y^2}{39} = 1 \). The video concludes with an invitation for viewers to ask questions or seek clarifications in the comments section.

Takeaways
  • 📚 The video is a tutorial on finding the equation of an ellipse given its vertices and foci.
  • 📍 The vertices provided are (±8, 0) and (±5, 0), indicating the major and minor axes of the ellipse.
  • 🔍 The major axis is horizontal, which means the ellipse is oriented along the x-axis.
  • 📐 The standard form of the ellipse equation is \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), where \( a \) is the semi-major axis and \( b \) is the semi-minor axis.
  • 🧩 The semi-major axis \( a \) is determined to be 8 units, as it is the distance from the center to a vertex along the x-axis.
  • 🔢 The formula for the semi-minor axis \( b \) is derived from \( c = \sqrt{a^2 - b^2} \), where \( c \) is the distance from the center to a focus.
  • 🌐 The distance \( c \) is calculated using the given foci (±5, 0), leading to the equation \( 5 = \sqrt{8^2 - b^2} \).
  • 📉 Solving for \( b^2 \) gives \( b^2 = 64 - 25 = 39 \), so \( b = \sqrt{39} \).
  • 📝 The final equation of the ellipse is \( \frac{x^2}{64} + \frac{y^2}{39} = 1 \), using the calculated values for \( a^2 \) and \( b^2 \).
  • 👨‍🏫 The presenter is Prof D, who encourages viewers to ask questions or seek clarifications in the comment section.
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 given the vertices and the foci.

  • What are the vertices given in the video?

    -The vertices given in the video are (±8, 0) and (±5, 0).

  • What does the orientation of the vertices indicate about the major axis of the ellipse?

    -Since the vertices are aligned horizontally, it indicates that the major axis of the ellipse is horizontal.

  • What is the general form of the equation of an ellipse?

    -The general form of the equation of an ellipse is \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \).

  • How is the value of 'a' determined in the equation of an ellipse?

    -The value of 'a' is determined by the distance from the center of the ellipse to one of its vertices along the major axis.

  • What is the relationship between 'a', 'b', and 'c' in an ellipse?

    -In an ellipse, 'c' is the distance from the center to a focus, and it is calculated using the formula \( c = \sqrt{a^2 - b^2} \).

  • How do you find the value of 'b' in the equation of an ellipse?

    -The value of 'b' can be found using the relationship \( b^2 = a^2 - c^2 \).

  • What is the final equation of the ellipse in the video?

    -The final equation of the ellipse is \( \frac{x^2}{64} + \frac{y^2}{39} = 1 \).

  • What is the role of 'a' and 'b' in the final equation of the ellipse?

    -In the final equation, 'a' is the semi-major axis and 'b' is the semi-minor axis of the ellipse.

  • Who is the presenter of the video?

    -The presenter of the video is Prof D.

  • How can viewers ask questions or seek clarifications after watching the video?

    -Viewers can ask questions or seek clarifications by posting them in the comment section below the video.

Outlines
00:00
📐 Finding the Equation of an Ellipse

In this video, the instructor explains how to find the equation of an ellipse given its vertices and foci. The vertices are at (8, 0) and (-8, 0), and the foci are at (5, 0) and (-5, 0). The major axis is horizontal, indicating a horizontal ellipse. The formula used involves identifying the values of 'a' and 'b' from the vertices and foci. The steps include calculating 'a' and 'b', solving for 'b' using the equation c² = a² - b², and substituting the values back into the standard form of the ellipse equation. The final equation of the ellipse is x²/64 + y²/39 = 1.

05:01
🔔 Conclusion and Call for Questions

The instructor wraps up the video, expressing hope that the lesson was understood. Viewers are encouraged to leave any questions or clarifications in the comments section. The instructor signs off, thanking viewers for watching and promising to see them in the next video.

Mindmap
Keywords
💡Ellipse
An ellipse is a geometric shape that resembles a flattened circle. It is defined as the set of all points for which the sum of the distances to two fixed points, called the foci, is constant. In the context of the video, the ellipse is the main subject, and the process of finding its equation is the core educational content.
💡Vertices
Vertices in geometry refer to the points where the major axis of an ellipse intersects its boundary. For an ellipse, there are two pairs of vertices, one on each axis. In the video, the vertices are given as (±8, 0) and (0, ±5), which are used to determine the lengths of the semi-major and semi-minor axes.
💡Psi (ψ)
Psi (ψ) is not explicitly defined in the script, but it seems to be a placeholder or a typo for 'focus'. The foci of an ellipse are two points such that the sum of the distances from any point on the ellipse to the foci is constant. In the video, the term '4 psi' is likely meant to refer to the distance of the foci from the center along the minor axis.
💡Cartesian Plane
A Cartesian plane is a two-dimensional coordinate system where each point is defined by an ordered pair of numbers, typically (x, y). It is named after the French mathematician René Descartes. In the video, the Cartesian plane is the reference for plotting the vertices and foci of the ellipse.
💡Major Axis
The major axis of an ellipse is the longest diameter, which passes through both foci and the center of the ellipse. It is used to determine the semi-major axis length, denoted as 'a'. In the video, the major axis is horizontal, indicating that the ellipse is elongated along the x-axis.
💡Horizontal
Horizontal refers to a direction parallel to the x-axis in the Cartesian plane. The term is used in the video to describe the orientation of the major axis of the ellipse, which is crucial for determining the form of the ellipse's equation.
💡Equation of an Ellipse
The equation of an ellipse is a mathematical formula that describes the set of points that form the ellipse. It is typically written in the form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), where 'a' is the semi-major axis and 'b' is the semi-minor axis. The video's main goal is to derive this equation given the vertices and foci of the ellipse.
💡Semi-major Axis (a)
The semi-major axis (denoted as 'a') is half the length of the major axis of an ellipse. It is used in the equation of the ellipse to determine the x-coordinates of the ellipse's boundary. In the video, 'a' is calculated using the distance from the center to the vertices along the x-axis.
💡Semi-minor Axis (b)
The semi-minor axis (denoted as 'b') is half the length of the minor axis of an ellipse, which is perpendicular to the major axis. It is used in the equation of the ellipse to determine the y-coordinates of the ellipse's boundary. In the video, 'b' is calculated using the distance from the center to the vertices along the y-axis.
💡Foci
Foci (plural of focus) are two points inside an ellipse such that the sum of the distances from any point on the ellipse to the foci is constant. The foci are crucial for defining the shape of the ellipse. In the script, the foci are given as (±5, 0), indicating their positions relative to the center of the ellipse.
💡Formula
In mathematics, a formula is a concise way of expressing information symbolically as a mathematical sentence, which can include symbols, numbers, and words. In the video, several formulas are used to calculate the lengths of the semi-major and semi-minor axes and to derive the final equation of the ellipse.
Highlights

Introduction to finding the equation of an ellipse given vertices and psi.

Vertices are provided as positive and negative eight, zero and four.

Focus is identified as being on a horizontal line, indicating a horizontal major axis.

The equation of a horizontally oriented ellipse is introduced.

Formula for identifying the semi-major axis 'a' from vertices is explained.

The relationship between 'a', 'b', and 'c' in an ellipse is discussed.

Calculation of 'c' using the formula c = sqrt(a^2 - b^2) is shown.

Substitution of given values to find 'b' is demonstrated.

Solving for 'b' by rearranging the equation to isolate b^2.

The value of b^2 is calculated to be 39.

The semi-minor axis 'b' is found to be the square root of 39.

The general equation of the ellipse is applied with found values of 'a' and 'b'.

The final equation of the ellipse is presented as (x^2/64) + (y^2/39) = 1.

The video concludes with a summary of the ellipse's equation.

An invitation for questions or clarifications is extended to the viewers.

The video ends with a sign-off from Prof D.

Transcripts

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Ellipse EquationVerticesPsiMathematicsEducationalTutorialCartesian PlaneMajor AxisMinor AxisProf D