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Grade 11 · lesson 3 of 4

Distinguish a Semi-Axis from a Full Axis

For an origin-centered ellipse with positive squared denominators, the larger denominator supplies the squared semi-major axis.

About 25 minutes, plus any extended practice

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Supplied materials

  • Paper or a browser scratch area; a calculator is optional unless rounding is requested.
  • Supplied teaching case: For x²/25+y²/4=1, what is the full major-axis length? Candidate answers: (A) 10 units (B) 25 units (C) 4 units (D) 5 units.
  • Independent check inputs: For x²/36+y²/9=1, what is the full major-axis length? Candidate answers: (A) 12 units (B) 14 units (C) 36 units (D) 6 units.

Learn the idea

Circle, parabola, ellipse and hyperbola equations encode geometric features through squared quantities and sign patterns. Standard forms locate centers, focal information, axes or vertices, with care to distinguish full lengths from half lengths. For an origin-centered ellipse with positive squared denominators, the larger denominator supplies the squared semi-major axis. Twice its positive root is the full major-axis length.

Here is the method on the supplied teaching case. The larger denominator is the square of the horizontal semi-major axis. Its half-length is 5; the full length is 2×5=10.

The denominator is a squared half-length, not the full axis length. Identify which variable has the larger denominator to determine the major-axis direction before reporting a length.

  1. Write or sketch the supplied teaching case, labeling its quantities and restrictions.
  2. Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
  3. Solve the independent check with its changed inputs before revealing feedback.
  4. Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.
Worked example

See the method in action

Teaching case: For x²/25+y²/4=1, what is the full major-axis length? Candidate answers: (A) 10 units (B) 25 units (C) 4 units (D) 5 units. Solution: The larger denominator is the square of the horizontal semi-major axis. Its half-length is 5; the full length is 2×5=10. The final answer is 10 units. The denominator is a squared half-length, not the full axis length. Identify which variable has the larger denominator to determine the major-axis direction before reporting a length.

Your turn

Check your reasoning

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Helpful tips
  • For an origin-centered ellipse with positive squared denominators, the larger denominator supplies the squared semi-major axis.
  • The denominator is a squared half-length, not the full axis length.
For x²/36+y²/9=1, what is the full major-axis length?

Feedback appears here after you check. This is practice, not a graded school assessment.

See the answer and explanation

12 units

The larger denominator is the square of the horizontal semi-major axis. Its half-length is 6; the full length is 2×6=12.

Take it further

Changed-input task: For x²/144+y²/81=1, what is the full major-axis length? Candidate answers: (A) 12 units (B) 144 units (C) 18 units (D) 24 units. Show the mathematical reason, not only the selected answer. Expected result: 24 units. The larger denominator is the square of the horizontal semi-major axis. Its half-length is 12; the full length is 2×12=24.

Nothing here is saved to an account or sent to a teacher. The answer download includes your current choice and any checked feedback. It does not include choices from the separate More practice section. Download that section separately.

More practice: try new cases

Choose an answer for each problem. Explain your method on paper, then use the feedback to decide what to revisit.

More practice: try new cases
1. For x²/49+y²/16=1, what is the full major-axis length?
Need a hint?
  • For an origin-centered ellipse with positive squared denominators, the larger denominator supplies the squared semi-major axis.
  • The denominator is a squared half-length, not the full axis length.
2. For x²/64+y²/25=1, what is the full major-axis length?
Need a hint?
  • For an origin-centered ellipse with positive squared denominators, the larger denominator supplies the squared semi-major axis.
  • The denominator is a squared half-length, not the full axis length.
3. For x²/81+y²/36=1, what is the full major-axis length?
Need a hint?
  • For an origin-centered ellipse with positive squared denominators, the larger denominator supplies the squared semi-major axis.
  • The denominator is a squared half-length, not the full axis length.

Your choices are temporary. Feedback appears after you check all the answers.

Nothing here is saved to an account or sent to a teacher. Keep your own notes or download them before leaving. The score describes these questions; it does not certify subject mastery.

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These LifeStarter passages, scenarios and exercises are authored for this Academy with AI assistance and review. External resources keep their publishers’ terms. A grade label suggests a starting level; it does not establish alignment with your school’s complete requirements.