Unlocking Ellipses: Mastering Equations from Vertices and Foci

Algebra 2 Grades High School 4:50 Video
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Lesson Description

Learn how to derive the equation of an ellipse when given its vertices and foci. This lesson covers the fundamental concepts, formulas, and a step-by-step example to enhance your understanding of ellipses.

Video Resource

Equation of Ellipse Given Vertices and Foci

Mario's Math Tutoring

Duration: 4:50
Watch on YouTube

Key Concepts

  • Ellipse definition and properties (vertices, co-vertices, foci, center)
  • Standard equation of an ellipse (horizontal and vertical major axes)
  • Relationship between a, b, and c (c² = a² - b²)
  • Eccentricity of an ellipse

Learning Objectives

  • Students will be able to identify the vertices, co-vertices, foci, and center of an ellipse.
  • Students will be able to determine the equation of an ellipse given its vertices and foci.
  • Students will be able to apply the relationship c² = a² - b² to find missing parameters of an ellipse.
  • Students will be able to calculate the eccentricity of an ellipse given its foci and vertices.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the definition of an ellipse and its key components: vertices, co-vertices, foci, and center. Briefly discuss the general equation of an ellipse and how the values of 'a' and 'b' relate to the major and minor axes.
  • Equation of an Ellipse (10 mins)
    Explain the standard form of the ellipse equation: (x-h)²/a² + (y-k)²/b² = 1 (horizontal major axis) and (x-h)²/b² + (y-k)²/a² = 1 (vertical major axis). Emphasize the importance of 'a' being the distance from the center to the vertex and 'b' being the distance from the center to the co-vertex. Discuss how (h, k) represents the center of the ellipse and how to find the center given the coordinates of the vertices.
  • Relationship Between a, b, and c (5 mins)
    Introduce the equation c² = a² - b², where 'c' is the distance from the center to the focus. Explain how this equation is used to find 'b' when 'a' and 'c' are known, which is crucial for determining the complete equation of the ellipse.
  • Formula for Eccentricity (5 mins)
    Introduce the concept of eccentricity (e = c/a). Explain that eccentricity is a measure of how 'stretched' an ellipse is. An eccentricity of 0 represents a circle, while values closer to 1 represent more elongated ellipses.
  • Example Problem (15 mins)
    Work through the example provided in the video: Find the equation of the ellipse with vertices (-1,-3), (-7,-3) and foci (-2,-3), (-6,-3). Follow Mario's steps: 1) Sketch the points to visualize the ellipse. 2) Find the center by finding the midpoint of the vertices. 3) Determine 'a' as the distance from the center to a vertex. 4) Determine 'c' as the distance from the center to a focus. 5) Use c² = a² - b² to find 'b²'. 6) Plug the values of (h, k), a², and b² into the appropriate standard equation of the ellipse.
  • Practice Problems (10 mins)
    Provide students with practice problems where they are given the vertices and foci and asked to find the equation of the ellipse. Encourage them to follow the same steps as in the example problem.

Interactive Exercises

  • Graphing Tool Exploration
    Use online graphing tools (e.g., Desmos, GeoGebra) to graph ellipses given their equations. Students can manipulate the values of 'a', 'b', 'h', and 'k' and observe how these changes affect the ellipse's shape and position. They can also input vertices and foci and try to determine the equation that matches the ellipse.

Discussion Questions

  • How does the location of the vertices and foci determine whether the major axis is horizontal or vertical?
  • What is the significance of the values of 'a', 'b', and 'c' in determining the shape and size of the ellipse?
  • How does eccentricity relate to the shape of the ellipse? What does an eccentricity close to 0 vs. close to 1 mean?

Skills Developed

  • Analytical skills (analyzing given information to determine ellipse parameters)
  • Problem-solving skills (applying formulas and procedures to find the equation of an ellipse)
  • Visual reasoning (sketching graphs to aid in problem-solving)

Multiple Choice Questions

Question 1:

The distance from the center of an ellipse to a vertex is denoted by which variable?

Correct Answer: a

Question 2:

In the equation of an ellipse, (x-h)²/a² + (y-k)²/b² = 1, what does (h, k) represent?

Correct Answer: The center

Question 3:

Which equation relates a, b, and c in an ellipse?

Correct Answer: c² = a² - b²

Question 4:

If the vertices of an ellipse are (-3, 2) and (5, 2), what is the x-coordinate of the center?

Correct Answer: 1

Question 5:

The foci of an ellipse are always located on the:

Correct Answer: Major axis

Question 6:

What does the eccentricity of an ellipse measure?

Correct Answer: How stretched the ellipse is

Question 7:

If a = 5 and c = 3 for an ellipse, what is the value of b²?

Correct Answer: 16

Question 8:

In the standard equation of an ellipse, if the larger denominator is under the y² term, the major axis is:

Correct Answer: Diagonal

Question 9:

Which of the following values of eccentricity would represent a circle?

Correct Answer: e = 0

Question 10:

The eccentricity of an ellipse is defined as:

Correct Answer: c/a

Fill in the Blank Questions

Question 1:

The distance from the center of an ellipse to a focus is denoted by the variable ____.

Correct Answer: c

Question 2:

The midpoint between the vertices of an ellipse is the _____ of the ellipse.

Correct Answer: center

Question 3:

If the major axis of an ellipse is horizontal, the larger denominator in the standard equation will be under the _____ term.

Correct Answer: x²

Question 4:

The equation that relates a, b, and c in an ellipse is c² = a² - _____.

Correct Answer: b²

Question 5:

The distance from the center to a co-vertex is represented by the variable _____.

Correct Answer: b

Question 6:

Eccentricity is calculated by the formula: e = _____ / a

Correct Answer: c

Question 7:

An ellipse with an eccentricity close to _____ is nearly circular.

Correct Answer: 0

Question 8:

If the vertices are at (0, 5) and (0, -5), the length of the major axis is _____.

Correct Answer: 10

Question 9:

If a = 4 and b = 3, then c² = _____.

Correct Answer: 7

Question 10:

The distance from the center of an ellipse to either vertex is also known as the _____.

Correct Answer: semi-major axis

Educational Standards

A2.A.1:Represent and solve mathematical and real-world problems using nonlinear equations, systems of linear equations, and systems of linear inequalities; interpret the solutions in the original context. A2.G.3:Know and apply the distance formula and the midpoint formula, and derive the equation of a circle in the coordinate plane.

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