Kenneth ChengMathematical modeling
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LEARN · EXPLAIN · REVISE

Ellipse

Read the idea, work independently, then explain what changed.

高二選擇性必修 第一册(A版).pdf · 3.1 · PDF 110 / printed page 105

Revisit first: Equations of a circle

TOPIC 01

Ellipse

Use the focal definition, standard equation and geometric properties of an ellipse.

What you will be able to explain

  • Use the focal definition, standard equation and geometric properties of an ellipse.
  • Justify the method and check the conditions in a new situation.

Focal sum

Distances to the two foci sum to 2a; c²=a²−b².

x2/a2+y2/b2=1x^2/a^2+y^2/b^2=1

Axis conditions

For a horizontal major axis a>b>0; the larger denominator specifies the major axis.

PREDICT → EXPLORE → EXPLAIN → TRANSFER

Make a prediction, then explore the relationship.

Lesson question: Predict the ellipse shape and eccentricity as b approaches a or zero.

Model exploration: predict which displayed result changes with the parameters, check the values, and compare the observation with the lesson question.

Ellipse with semiaxes 2,1. A circle occurs when a=b; focus distance=1.7321.

Ellipse with semiaxes 2,1. A circle occurs when a=b; focus distance=1.7321.

Explain: Calculate two valid cases and explain the change using the defining relation.

Transfer: Compare the limiting geometry with the conditions for a nondegenerate ellipse.

Try it. Leave your reasoning visible.

Use one hint at a time. A correction explains what changed, not just the final answer.

01 / Foundation#Worked example

Find the major-axis length.

x2/16+y2/4=1x^2/16+y^2/4=1
  • For a horizontal major axis a>b>0; the larger denominator specifies the major axis.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Translate the geometric condition into a coordinate equation.
Hint 2
Use this intermediate relation.
major axis=2a\text{major axis}=2a
Worked solution

  1. Translate the geometric condition into a coordinate equation.

  2. Apply the stated relation and retain its conditions.

    2a=2(4)=82a=2(4)=8

  3. An axis is twice the corresponding semiaxis.

The requested value is 8.

Checks and common pitfalls: An axis is twice the corresponding semiaxis.

Think first. Reveal a hint when the class is ready.

02 / Standard#Worked example

Find the horizontal ellipse with a=t+1 and c=t.

t=3t=3
  • For a horizontal major axis a>b>0; the larger denominator specifies the major axis.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Translate the geometric condition into a coordinate equation.
Hint 2
Use this intermediate relation.
b2=a2−c2b²=a²−c²
Worked solution

  1. Translate the geometric condition into a coordinate equation.

  2. Apply the stated relation and retain its conditions.

    b2=16−9=7b^2=16-9=7

  3. The positive difference guarantees a nondegenerate ellipse.

The requested relation or conclusion is shown below.

x2/16+y2/7=1x^2/16+y^2/7=1

Checks and common pitfalls: The positive difference guarantees a nondegenerate ellipse.

Reasoning checklist · self / teacher assessment
  • State a valid definition or model and its assumptions.
  • Show the intermediate mathematical relations, not only the final claim.
  • Check exclusions, units or the interpretation of the result.

Think first. Reveal a hint when the class is ready.

03 / Transfer#Worked example

Find the largest value of 3x+4y on the ellipse.

x2/36+y2/16=1x^2/36+y^2/16=1
  • For a horizontal major axis a>b>0; the larger denominator specifies the major axis.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Translate the geometric condition into a coordinate equation.
Hint 2
Use this intermediate relation.
x=acosu,y=bsinux=a cos u, y=b sin u
Worked solution

  1. Translate the geometric condition into a coordinate equation.

  2. Apply the stated relation and retain its conditions.

    3x+4y=18cos⁡u+16sin⁡u3x+4y=18\cos u+16\sin u

  3. Apply the stated relation and retain its conditions.

    max⁡=580\max=\sqrt{580}

  4. The full trigonometric range attains the amplitude.

The requested value is 24.0831891576.

Checks and common pitfalls: The full trigonometric range attains the amplitude.

Think first. Reveal a hint when the class is ready.

04 / Foundation#Your turn

Find the major-axis length.

x2/49+y2/25=1x^2/49+y^2/25=1
  • For a horizontal major axis a>b>0; the larger denominator specifies the major axis.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Translate the geometric condition into a coordinate equation.
Hint 2
Use this intermediate relation.
major axis=2a\text{major axis}=2a
Worked solution

  1. Translate the geometric condition into a coordinate equation.

  2. Apply the stated relation and retain its conditions.

    2a=2(7)=142a=2(7)=14

  3. An axis is twice the corresponding semiaxis.

The requested value is 14.

Checks and common pitfalls: An axis is twice the corresponding semiaxis.

Think first. Reveal a hint when the class is ready.

05 / Foundation#Your turn

Find the squared focal distance c².

a=8,b=6a=8,\quad b=6
  • For a horizontal major axis a>b>0; the larger denominator specifies the major axis.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Translate the geometric condition into a coordinate equation.
Hint 2
Use this intermediate relation.
c2=a2−b2c^2=a^2-b^2
Worked solution

  1. Translate the geometric condition into a coordinate equation.

  2. Apply the stated relation and retain its conditions.

    c2=64−36=28c^2=64-36=28

  3. For an ellipse, subtract the squared semiaxes.

The requested value is 28.

Checks and common pitfalls: For an ellipse, subtract the squared semiaxes.

Think first. Reveal a hint when the class is ready.

06 / Foundation#Your turn

Find the eccentricity.

x2/81+y2/49=1x^2/81+y^2/49=1
  • For a horizontal major axis a>b>0; the larger denominator specifies the major axis.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Translate the geometric condition into a coordinate equation.
Hint 2
Use this intermediate relation.
e=c/ae=c/a
Worked solution

  1. Translate the geometric condition into a coordinate equation.

  2. Apply the stated relation and retain its conditions.

    c=32c=\sqrt{32}

  3. Apply the stated relation and retain its conditions.

    e=32/9e=\sqrt{32}/9

  4. A noncircular nondegenerate ellipse has 0<e<1.

The requested value is 0.628539361055.

Checks and common pitfalls: A noncircular nondegenerate ellipse has 0<e<1.

Think first. Reveal a hint when the class is ready.

07 / Foundation#Your turn

P is on the ellipse and PF1=t. Find PF2.

a=10,PF1=8a=10,\quad PF_1=8
  • For a horizontal major axis a>b>0; the larger denominator specifies the major axis.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Translate the geometric condition into a coordinate equation.
Hint 2
Use this intermediate relation.
PF1+PF2=2aPF_1+PF_2=2a
Worked solution

  1. Translate the geometric condition into a coordinate equation.

  2. Apply the stated relation and retain its conditions.

    PF2=20−8=12PF_2=20-8=12

  3. The distance-sum definition avoids solving coordinates.

The requested value is 12.

Checks and common pitfalls: The distance-sum definition avoids solving coordinates.

Think first. Reveal a hint when the class is ready.

08 / Standard#Your turn

P is on the ellipse and PF1=t. Find PF2.

a=11,PF1=9a=11,\quad PF_1=9
  • For a horizontal major axis a>b>0; the larger denominator specifies the major axis.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Translate the geometric condition into a coordinate equation.
Hint 2
Use this intermediate relation.
PF1+PF2=2aPF_1+PF_2=2a
Worked solution

  1. Translate the geometric condition into a coordinate equation.

  2. Apply the stated relation and retain its conditions.

    PF2=22−9=13PF_2=22-9=13

  3. The distance-sum definition avoids solving coordinates.

The requested value is 13.

Checks and common pitfalls: The distance-sum definition avoids solving coordinates.

Think first. Reveal a hint when the class is ready.

09 / Standard#Your turn

Find the horizontal ellipse with a=t+1 and c=t.

t=10t=10
  • For a horizontal major axis a>b>0; the larger denominator specifies the major axis.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Translate the geometric condition into a coordinate equation.
Hint 2
Use this intermediate relation.
b2=a2−c2b²=a²−c²
Worked solution

  1. Translate the geometric condition into a coordinate equation.

  2. Apply the stated relation and retain its conditions.

    b2=121−100=21b^2=121-100=21

  3. The positive difference guarantees a nondegenerate ellipse.

The requested relation or conclusion is shown below.

x2/121+y2/21=1x^2/121+y^2/21=1

Checks and common pitfalls: The positive difference guarantees a nondegenerate ellipse.

Reasoning checklist · self / teacher assessment
  • State a valid definition or model and its assumptions.
  • Show the intermediate mathematical relations, not only the final claim.
  • Check exclusions, units or the interpretation of the result.

Think first. Reveal a hint when the class is ready.

10 / Standard#Your turn

Find the length of the chord cut by x=0.

x2/169+y2/121=1x^2/169+y^2/121=1
  • For a horizontal major axis a>b>0; the larger denominator specifies the major axis.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Translate the geometric condition into a coordinate equation.
Hint 2
Use this intermediate relation.
y2/b2=1y^2/b^2=1
Worked solution

  1. Translate the geometric condition into a coordinate equation.

  2. Apply the stated relation and retain its conditions.

    y=±11⇒L=22y=\pm11\Rightarrow L=22

  3. The chord through the center along the y-axis is the minor axis.

The requested value is 22.

Checks and common pitfalls: The chord through the center along the y-axis is the minor axis.

Think first. Reveal a hint when the class is ready.

11 / Standard#Your turn

Find the largest value of 3x+4y on the ellipse.

x2/196+y2/144=1x^2/196+y^2/144=1
  • For a horizontal major axis a>b>0; the larger denominator specifies the major axis.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Translate the geometric condition into a coordinate equation.
Hint 2
Use this intermediate relation.
x=acosu,y=bsinux=a cos u, y=b sin u
Worked solution

  1. Translate the geometric condition into a coordinate equation.

  2. Apply the stated relation and retain its conditions.

    3x+4y=42cos⁡u+48sin⁡u3x+4y=42\cos u+48\sin u

  3. Apply the stated relation and retain its conditions.

    max⁡=4068\max=\sqrt{4068}

  4. The full trigonometric range attains the amplitude.

The requested value is 63.7808748764.

Checks and common pitfalls: The full trigonometric range attains the amplitude.

Think first. Reveal a hint when the class is ready.

12 / Transfer#Your turn

Find the length of the chord cut by x=0.

x2/225+y2/169=1x^2/225+y^2/169=1
  • For a horizontal major axis a>b>0; the larger denominator specifies the major axis.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Translate the geometric condition into a coordinate equation.
Hint 2
Use this intermediate relation.
y2/b2=1y^2/b^2=1
Worked solution

  1. Translate the geometric condition into a coordinate equation.

  2. Apply the stated relation and retain its conditions.

    y=±13⇒L=26y=\pm13\Rightarrow L=26

  3. The chord through the center along the y-axis is the minor axis.

The requested value is 26.

Checks and common pitfalls: The chord through the center along the y-axis is the minor axis.

Think first. Reveal a hint when the class is ready.

13 / Transfer#Your turn

Find the largest value of 3x+4y on the ellipse.

x2/256+y2/196=1x^2/256+y^2/196=1
  • For a horizontal major axis a>b>0; the larger denominator specifies the major axis.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Translate the geometric condition into a coordinate equation.
Hint 2
Use this intermediate relation.
x=acosu,y=bsinux=a cos u, y=b sin u
Worked solution

  1. Translate the geometric condition into a coordinate equation.

  2. Apply the stated relation and retain its conditions.

    3x+4y=48cos⁡u+56sin⁡u3x+4y=48\cos u+56\sin u

  3. Apply the stated relation and retain its conditions.

    max⁡=5440\max=\sqrt{5440}

  4. The full trigonometric range attains the amplitude.

The requested value is 73.7563556583.

Checks and common pitfalls: The full trigonometric range attains the amplitude.

Think first. Reveal a hint when the class is ready.

Focus on one question

End-of-lesson check and correction

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    Choose a foundation skill to revisit ↗

    Teacher preparation and assessment

    Question sequence

    • Use the focal definition, standard equation and geometric properties of an ellipse.
    • Which condition is essential in ellipse?
    • Predict the ellipse shape and eccentricity as b approaches a or zero.

    Board plan

    • Focal sum: Distances to the two foci sum to 2a; c²=a²−b².
      x2/a2+y2/b2=1x^2/a^2+y^2/b^2=1
    • Axis conditions: For a horizontal major axis a>b>0; the larger denominator specifies the major axis.

    Anticipated thinking

    • The letter attached to x² is not always the major semiaxis.

    Assessment checklist

    • 1 mark: choose the correct representation and conditions.
    • 1 mark: establish the intermediate relation.
    • 1 mark: complete a connected calculation or proof.
    • 1 mark: interpret and check the conclusion.

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    Curriculum and source notes ↗