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Conic sections practice

Properties of ellipses worksheets

Find exact centers, vertices, co-vertices, foci, axis lengths and eccentricity from standard-form ellipse equations.

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Identify the requested exact properties of each ellipse.

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US Letter

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A4

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Read every exact feature from standard form

Each worksheet contains four ellipses. Choose any nonempty set of center, vertices, co-vertices, foci, axis lengths and eccentricity.

Compare the squared semiaxis denominators to find the major direction, then use c² = a² − b². Keep translated radical coordinates exact.

The separate key supplies only the properties selected for the student sheet.

Learning objective

What this activity builds

Derive exact geometric properties from an axis-aligned ellipse in standard form.

Worked example

Find the foci of x²/25 + y²/9 = 1.

(−4, 0) and (4, 0).

Here a² = 25, b² = 9 and c² = 16, so c = 4 along the horizontal major axis.

Teaching notes

Make the practice count

  • Compare the two denominators to identify the major direction.
  • Use c² = a² − b² for the focal distance.
  • Keep translated radical coordinates exact.
  • Axis lengths are twice their semiaxis lengths.

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Choose one or more exact properties, then change the seed for four reproducible ellipses.

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Math SheetNameScore/ 28TeacherDateProperties of EllipsesIdentify the requested exact properties of each ellipse.1.x2121+y-4236=1Center: ______________________________Vertices: ______________________________Co-vertices: ______________________________Foci: ______________________________Major axis length: ______________________________Minor axis length: ______________________________Eccentricity: ______________________________2.x-629+y2100=1Center: ______________________________Vertices: ______________________________Co-vertices: ______________________________Foci: ______________________________Major axis length: ______________________________Minor axis length: ______________________________Eccentricity: ______________________________3.x+124+y+121=1Center: ______________________________Vertices: ______________________________Co-vertices: ______________________________Foci: ______________________________Major axis length: ______________________________Minor axis length: ______________________________Eccentricity: ______________________________4.x+221+y+529=1Center: ______________________________Vertices: ______________________________Co-vertices: ______________________________Foci: ______________________________Major axis length: ______________________________Minor axis length: ______________________________Eccentricity: ______________________________math-sheet.comPage 1 / 1

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