Kenneth ChengMathematical modeling
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LEARN · EXPLAIN · REVISE

Hyperbola

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

高二選擇性必修 第一册(A版).pdf · 3.2 · PDF 123 / printed page 118

Revisit first: Ellipse

TOPIC 01

Hyperbola

Use the distance-difference definition, asymptotes and branch restrictions.

What you will be able to explain

  • Use the distance-difference definition, asymptotes and branch restrictions.
  • Justify the method and check the conditions in a new situation.

Parameters

For x²/a²−y²/b²=1, c²=a²+b² and asymptotes are y=±(b/a)x.

Branch conditions

a,b>0 and e>1; a distance difference is absolute, and an asymptote is not part of the curve.

PREDICT → EXPLORE → EXPLAIN → TRANSFER

Make a prediction, then explore the relationship.

Lesson question: Can a line have one intersection with a hyperbola without being a tangent?

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

Hyperbola x²/4−y²/1=1; a,b>0.

Hyperbola x²/4−y²/1=1; a,b>0.

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

Transfer: Compare a tangent with a line parallel to an asymptote and explain the reduced equation.

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 c².

x2/4−y2/4=1x^2/4-y^2/4=1
  • a,b>0 and e>1; a distance difference is absolute, and an asymptote is not part of the curve.
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=4+4=8c^2=4+4=8

  3. The focal square is a sum for a hyperbola.

The requested value is 8.

Checks and common pitfalls: The focal square is a sum for a hyperbola.

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

02 / Standard#Worked example

Find the horizontal hyperbola with vertices (±t,0) and foci (±(t+2),0).

t=3t=3
  • a,b>0 and e>1; a distance difference is absolute, and an asymptote is not part of the curve.
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=c2−a2b²=c²−a²
Worked solution

  1. Translate the geometric condition into a coordinate equation.

  2. Apply the stated relation and retain its conditions.

    b2=25−9=16b^2=25-9=16

  3. For a hyperbola c exceeds a.

The requested relation or conclusion is shown below.

x2/9−y2/16=1x^2/9-y^2/16=1

Checks and common pitfalls: For a hyperbola c exceeds a.

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

Show there is one common point and explain why this line is not tangent.

y=x+4,x2−y2=16y=x+4, x^2-y^2=16
  • a,b>0 and e>1; a distance difference is absolute, and an asymptote is not part of the curve.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Translate the geometric condition into a coordinate equation.
Hint 2
Substitution cancels the quadratic terms.
Worked solution

  1. Translate the geometric condition into a coordinate equation.

  2. Apply the stated relation and retain its conditions.

    −8x−16=16⇒x=−4,y=0-8x-16=16 ⇒ x=-4, y=0

  3. Apply the stated relation and retain its conditions.

    tangent at P:x=−4\text{tangent at }P:x=-4

  4. The line is parallel to an asymptote; one intersection can result from degree reduction.

The requested relation or conclusion is shown below.

P=(−4,0)P=(-4,0)

Checks and common pitfalls: The line is parallel to an asymptote; one intersection can result from degree reduction.

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.

04 / Foundation#Your turn

Find c².

x2/25−y2/4=1x^2/25-y^2/4=1
  • a,b>0 and e>1; a distance difference is absolute, and an asymptote is not part of the curve.
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=25+4=29c^2=25+4=29

  3. The focal square is a sum for a hyperbola.

The requested value is 29.

Checks and common pitfalls: The focal square is a sum for a hyperbola.

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

05 / Foundation#Your turn

Find the length of the transverse axis.

x2/36−y2/4=1x^2/36-y^2/4=1
  • a,b>0 and e>1; a distance difference is absolute, and an asymptote is not part of the curve.
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.
transverse axis=2a\text{transverse axis}=2a
Worked solution

  1. Translate the geometric condition into a coordinate equation.

  2. Apply the stated relation and retain its conditions.

    2a=122a=12

  3. Only the positive squared term specifies the transverse axis.

The requested value is 12.

Checks and common pitfalls: Only the positive squared term specifies the transverse axis.

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

06 / Foundation#Your turn

Find the positive asymptote slope.

x2/49−y2/4=1x^2/49-y^2/4=1
  • a,b>0 and e>1; a distance difference is absolute, and an asymptote is not part of the curve.
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.
x2/a2−y2/b2=0x^2/a^2-y^2/b^2=0
Worked solution

  1. Translate the geometric condition into a coordinate equation.

  2. Apply the stated relation and retain its conditions.

    y=±(2/7)xy=\pm(2/7)x

  3. The asymptotes come from the homogeneous quadratic part.

The requested value is 0.285714285714.

Checks and common pitfalls: The asymptotes come from the homogeneous quadratic part.

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

07 / Foundation#Your turn

Find the eccentricity.

a=8,b=2a=8,\quad b=2
  • a,b>0 and e>1; a distance difference is absolute, and an asymptote is not part of the curve.
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.

    e=68/8e=\sqrt{68}/8

  3. The computed value must exceed one.

The requested value is 1.0307764064.

Checks and common pitfalls: The computed value must exceed one.

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

08 / Standard#Your turn

Find the eccentricity.

a=9,b=2a=9,\quad b=2
  • a,b>0 and e>1; a distance difference is absolute, and an asymptote is not part of the curve.
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.

    e=85/9e=\sqrt{85}/9

  3. The computed value must exceed one.

The requested value is 1.02439382859.

Checks and common pitfalls: The computed value must exceed one.

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

09 / Standard#Your turn

Find the horizontal hyperbola with vertices (±t,0) and foci (±(t+2),0).

t=10t=10
  • a,b>0 and e>1; a distance difference is absolute, and an asymptote is not part of the curve.
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=c2−a2b²=c²−a²
Worked solution

  1. Translate the geometric condition into a coordinate equation.

  2. Apply the stated relation and retain its conditions.

    b2=144−100=44b^2=144-100=44

  3. For a hyperbola c exceeds a.

The requested relation or conclusion is shown below.

x2/100−y2/44=1x^2/100-y^2/44=1

Checks and common pitfalls: For a hyperbola c exceeds a.

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 intersection with x=0.

x2/121−y2/4=1x^2/121-y^2/4=1
  • a,b>0 and e>1; a distance difference is absolute, and an asymptote is not part of the curve.
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.
−y2/4=1-y^2/4=1
Worked solution

  1. Translate the geometric condition into a coordinate equation.

  2. Apply the stated relation and retain its conditions.

    y2=−4y^2=-4

  3. No real point can have a negative square; the conjugate-axis line does not meet this hyperbola.

The requested relation or conclusion is shown below.

∅\varnothing

Checks and common pitfalls: No real point can have a negative square; the conjugate-axis line does not meet this hyperbola.

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.

11 / Standard#Your turn

Show there is one common point and explain why this line is not tangent.

y=x+12,x2−y2=144y=x+12, x^2-y^2=144
  • a,b>0 and e>1; a distance difference is absolute, and an asymptote is not part of the curve.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Translate the geometric condition into a coordinate equation.
Hint 2
Substitution cancels the quadratic terms.
Worked solution

  1. Translate the geometric condition into a coordinate equation.

  2. Apply the stated relation and retain its conditions.

    −24x−144=144⇒x=−12,y=0-24x-144=144 ⇒ x=-12, y=0

  3. Apply the stated relation and retain its conditions.

    tangent at P:x=−12\text{tangent at }P:x=-12

  4. The line is parallel to an asymptote; one intersection can result from degree reduction.

The requested relation or conclusion is shown below.

P=(−12,0)P=(-12,0)

Checks and common pitfalls: The line is parallel to an asymptote; one intersection can result from degree reduction.

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.

12 / Transfer#Your turn

Find the intersection with x=0.

x2/169−y2/4=1x^2/169-y^2/4=1
  • a,b>0 and e>1; a distance difference is absolute, and an asymptote is not part of the curve.
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.
−y2/4=1-y^2/4=1
Worked solution

  1. Translate the geometric condition into a coordinate equation.

  2. Apply the stated relation and retain its conditions.

    y2=−4y^2=-4

  3. No real point can have a negative square; the conjugate-axis line does not meet this hyperbola.

The requested relation or conclusion is shown below.

∅\varnothing

Checks and common pitfalls: No real point can have a negative square; the conjugate-axis line does not meet this hyperbola.

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.

13 / Transfer#Your turn

Show there is one common point and explain why this line is not tangent.

y=x+14,x2−y2=196y=x+14, x^2-y^2=196
  • a,b>0 and e>1; a distance difference is absolute, and an asymptote is not part of the curve.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Translate the geometric condition into a coordinate equation.
Hint 2
Substitution cancels the quadratic terms.
Worked solution

  1. Translate the geometric condition into a coordinate equation.

  2. Apply the stated relation and retain its conditions.

    −28x−196=196⇒x=−14,y=0-28x-196=196 ⇒ x=-14, y=0

  3. Apply the stated relation and retain its conditions.

    tangent at P:x=−14\text{tangent at }P:x=-14

  4. The line is parallel to an asymptote; one intersection can result from degree reduction.

The requested relation or conclusion is shown below.

P=(−14,0)P=(-14,0)

Checks and common pitfalls: The line is parallel to an asymptote; one intersection can result from degree reduction.

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.

Focus on one question

End-of-lesson check and correction

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    Teacher preparation and assessment

    Question sequence

    • Use the distance-difference definition, asymptotes and branch restrictions.
    • Which condition is essential in hyperbola?
    • Can a line have one intersection with a hyperbola without being a tangent?

    Board plan

    • Parameters: For x²/a²−y²/b²=1, c²=a²+b² and asymptotes are y=±(b/a)x.
    • Branch conditions: a,b>0 and e>1; a distance difference is absolute, and an asymptote is not part of the curve.

    Anticipated thinking

    • Using c²=a²−b² confuses a hyperbola with an ellipse.

    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 ↗