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Questions and Answers
What is defined as a set of points in a plane whose sum of distances from two fixed points is constant?
Ellipse
What do we call the distance between the center and the focus of an ellipse?
Focal distance
What part of an ellipse is equal to the sum of the distances from point P to the two fixed points?
Major Axis
What is the line passing through the vertices and foci of an ellipse called?
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What are the two points on the ellipse that lie on the principal axis called?
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Determine the focal distance if the major axis and minor axis are 26 units and 10 units respectively.
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What is the line segment that is perpendicular to the major axis and connects the co-vertices of an ellipse called?
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What are the two standard forms of equations of an ellipse with the center at the origin?
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If the center of an ellipse is at the origin and the foci are at (0,8) and (0,-8), what is the form of the equation of the ellipse?
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Find the equation of an ellipse with a center at the origin, major axis 26 units and minor axis 10 units.
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The value of a is always greater than the value of b and c.
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If the values of a and c are given, what formula do you use to find the value of b?
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Find the equation of an ellipse with center at the origin, vertices at 0, ±10, and co-vertices at ±8, 0.
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Find the equation of an ellipse with center at (-1, 1), horizontal major axis, focal distance of 20 units and minor axis of length 42 units.
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Is the given equation x²/16 - y²/25 = 1 an equation of an ellipse?
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Is the given equation x²/36 + (y + 6)²/36 = 1 an equation of an ellipse?
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Determine the orientation of the major axis given the equation x²/36 + y²/49 = 1.
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Determine the focal distance given the equation of an ellipse x²/36 + y²/100 = 1.
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Complete the analogy: CIRCLE : A = B, ELLIPSE : A ____.
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What are the two standard forms of equations of an ellipse with center at (h,k)?
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Convert 3x + 4y - 24 = 0 into the standard form.
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Convert 5x + 7y - 35 = 0 into the standard form.
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What are the important points needed in graphing an ellipse?
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The distance between two vertices is the length of the major axis.
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Convert x²/6 + y²/7 = 1 into the general form.
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Convert x²/12 + y²/6 = 1 into the general form.
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In graphing an ellipse with a horizontal major axis, where do you plot the vertices from the center?
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In graphing an ellipse with a vertical major axis, where do you plot the co-vertices from the center?
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Study Notes
Ellipse Definition
- The sum of the distances from a point on the ellipse to two fixed points, called foci, is constant.
Focal Distance
- The distance between the center of an ellipse and a focus is known as the focal distance.
Major Axis
- The sum of the distances of a point on the ellipse to the two foci is equal to the length of the major axis.
Principal Axis
- The line passing through the vertices, foci, and center of an ellipse is called the principal axis.
Vertices
- The two points on the ellipse that lie on the principal axis are called the vertices.
Minor Axis
- The line segment perpendicular to the major axis and connecting the co-vertices of an ellipse is called the minor axis.
Standard Form Equations (Center at Origin)
- Horizontal Major Axis: 𝒙²/𝒂² + 𝒚²/𝒃² = 𝟏
- Vertical Major Axis: 𝒙²/𝒃² + 𝒚²/𝒂² = 𝟏
Finding the Equation of an Ellipse
- If the center is at the origin and the foci are at (0, 8) and (0, -8), the equation is 𝒙²/𝒃² + 𝒚²/𝒂² = 𝟏.
- If the center is at the origin, the major axis is vertical, and the major and minor axis lengths are 26 and 10 units respectively, the equation is 𝒙²/𝟐𝟓 + 𝒚²/𝟏𝟔𝟗 = 𝟏.
Ellipse Properties
- The value of "a" (half the length of the major axis) is always greater than the value of "b" (half the length of the minor axis) and "c" (focal distance).
- You can find the value of "b" using the following formula: 𝒃² = 𝒂² - 𝒄²
Finding the Equation from Vertices and Co-vertices
- If the center is at the origin, the vertices are at (0, ±10) and the co-vertices are at (±8, 0), the equation is 𝒙²/𝟔𝟒 + 𝒚²/𝟏𝟎𝟎 = 𝟏.
Finding the Equation from Center, Major Axis, and Focal Distance
- If the center is at (−1, 1), the horizontal major axis has a length of 42 units, and the focal distance is 20 units, the equation is (𝒙 + 𝟏)²/𝟖𝟒𝟏 + (𝒚 − 𝟏)²/𝟒𝟒𝟏 = 𝟏.
Identifying Ellipse Equations
- Not an Equation: If the operation between the terms containing 𝒙 and 𝒚 is subtraction, it is not an ellipse equation (example: (𝒙 − 2)²/𝟏𝟔 - 𝒚²/𝟐𝟓 = 𝟏).
- Equation: If the operation between the terms containing 𝒙 and 𝒚 is addition, it is an ellipse equation.
Determining Orientation
- If the denominator of the 𝒚² term is greater than the denominator of the 𝒙² term, the major axis is vertical (example: 𝒙²/𝟑𝟔 + 𝒚²/𝟒𝟗 = 𝟏).
- If the denominator of the 𝒙² term is greater than the denominator of the 𝒚² term, the major axis is horizontal.
Finding Focal Distance from the Equation
- Given the equation 𝒙²/𝟑𝟔 + 𝒚²/𝟏𝟎𝟎 = 𝟏, the focal distance (c) is 8 units, calculated using 𝒄² = 𝒂² - 𝒃² (where 𝒂 = 10 and 𝒃 = 6).
Ellipse vs Circle Analogy
- Circle: A = B (radius is constant)
- Ellipse: A ≠ B (major and minor axes are different)
Standard Form Equations (Center at (h, k))
- Horizontal Major Axis: (𝒙 − 𝒉)²/𝒂² + (𝒚 − 𝒌)²/𝒃² = 𝟏
- Vertical Major Axis: (𝒙 − 𝒉)²/𝒃² + (𝒚 − 𝒌)²/𝒂² = 𝟏
Converting General to Standard Form
- Convert 3𝒙² + 4𝒚² − 24 = 0 to the standard form: 𝒙²/𝟖 + 𝒚²/𝟔 = 𝟏.
- Convert 5𝒙² + 7𝒚² − 35 = 0 to the standard form: 𝒙²/𝟕 + 𝒚²/𝟓 = 𝟏.
Important Points for Graphing an Ellipse
- Center: (h, k)
- Foci: Foci are located a distance of "c" units from the center along the major axis.
- Vertices: Located at the ends of the major axis.
- Co-vertices: Located at the ends of the minor axis.
Relating Major Axis to Vertices
- The distance between the two vertices is equal to the length of the major axis.
Converting Standard Form to General Form
- Convert 𝒙²/𝟔 + 𝒚²/𝟕 = 𝟏 to the general form: 𝟕𝒙² + 𝟔𝒚² − 𝟒𝟐 = 𝟎.
- Convert 𝒙²/𝟏𝟐 + 𝒚²/𝟔 = 𝟏 to the general form: 𝒙² + 𝟐𝒚² − 𝟏𝟐 = 𝟎.
Graphing an Ellipse
- If the major axis is horizontal, plot the vertices to the right and left of the center.
- If the major axis is vertical, plot the co-vertices to the right and left of the center.
Act of Contrition
- This prayer is a form of confession of sins and a request for God's forgiveness.
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Description
Test your knowledge of the properties and equations of an ellipse with this quiz. Topics include focal distance, axes, vertices, and how to find the equation of an ellipse. Perfect for students studying conic sections in mathematics.
