Kenneth ChengMathematical modeling
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LEARN · EXPLAIN · REVISE

Relations between lines and circles

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

高二選擇性必修 第一册(A版).pdf · 2.5 · PDF 96 / printed page 91

Revisit first: Equations of a circle

TOPIC 01

Relations between lines and circles

Classify intersection and tangency using distances, discriminants and common chords.

What you will be able to explain

  • Classify intersection and tangency using distances, discriminants and common chords.
  • Justify the method and check the conditions in a new situation.

Line-circle relation

Compare center-to-line distance with radius; a secant chord has half-length √(r²−d²).

Circle-circle conditions

Compare center distance with r1+r2 and |r1−r2|; equal centers require separate treatment.

PREDICT → EXPLORE → EXPLAIN → TRANSFER

Make a prediction, then explore the relationship.

Lesson question: How many intersection points remain when a secant is moved away from the circle?

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

Circle x²+y²=4; line y=1. 2 intersections; chord length=3.4641. Tangency occurs exactly when h=r.

Circle x²+y²=4; line y=1. 2 intersections; chord length=3.4641. Tangency occurs exactly when h=r.

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

Transfer: Explain the same transition using distance and a quadratic discriminant.

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 number of common points.

x2+y2=4,y=2x^2+y^2=4,\quad y=2
  • Compare center distance with r1+r2 and |r1−r2|; equal centers require separate treatment.
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.
d=rd=r
Worked solution

  1. Translate the geometric condition into a coordinate equation.

  2. Apply the stated relation and retain its conditions.

    d=2=r⇒tangentd=2=r\Rightarrow\text{tangent}

  3. Equality gives one tangent point.

The requested value is 1.

Checks and common pitfalls: Equality gives one tangent point.

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

02 / Standard#Worked example

Find the radical axis and state when it is a common chord line.

x2+y2=25,(x−3)2+y2=25x^2+y^2=25,\quad(x-3)^2+y^2=25
  • Compare center distance with r1+r2 and |r1−r2|; equal centers require separate treatment.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Translate the geometric condition into a coordinate equation.
Hint 2
Subtract the equations.
Worked solution

  1. Translate the geometric condition into a coordinate equation.

  2. Apply the stated relation and retain its conditions.

    −2(3)x+9=0-2(3)x+9=0

  3. Apply the stated relation and retain its conditions.

    x=3/2x=3/2

  4. The center distance is 3. There are two common points, so this is their chord line.

The requested relation or conclusion is shown below.

x=1.5x=1.5

Checks and common pitfalls: The center distance is 3. There are two common points, so this is their chord line.

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 all values of a for which y=a is tangent to x²+y²=t².

t=4t=4
  • Compare center distance with r1+r2 and |r1−r2|; equal centers require separate treatment.
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.
d=∣a∣=rd=|a|=r
Worked solution

  1. Translate the geometric condition into a coordinate equation.

  2. Apply the stated relation and retain its conditions.

    ∣a∣=4⇒a=±4|a|=4\Rightarrow a=\pm4

  3. Both the upper and lower horizontal tangents must be retained.

The requested relation or conclusion is shown below.

a=±4a=\pm4

Checks and common pitfalls: Both the upper and lower horizontal tangents must be retained.

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 the number of common points.

x2+y2=25,y=5x^2+y^2=25,\quad y=5
  • Compare center distance with r1+r2 and |r1−r2|; equal centers require separate treatment.
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.
d=rd=r
Worked solution

  1. Translate the geometric condition into a coordinate equation.

  2. Apply the stated relation and retain its conditions.

    d=5=r⇒tangentd=5=r\Rightarrow\text{tangent}

  3. Equality gives one tangent point.

The requested value is 1.

Checks and common pitfalls: Equality gives one tangent point.

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

05 / Foundation#Your turn

Find the length of the chord.

x2+y2=25,y=3x^2+y^2=25,\quad y=3
  • Compare center distance with r1+r2 and |r1−r2|; equal centers require separate treatment.
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.
L=2r2−d2L=2\sqrt{r^2-d^2}
Worked solution

  1. Translate the geometric condition into a coordinate equation.

  2. Apply the stated relation and retain its conditions.

    L=225−9=8L=2\sqrt{25-9}=8

  3. The perpendicular from the center bisects the chord.

The requested value is 8.

Checks and common pitfalls: The perpendicular from the center bisects the chord.

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

06 / Foundation#Your turn

Find the tangent length from P to the circle.

P=(9,0),x2+y2=4P=(9,0),\quad x^2+y^2=4
  • Compare center distance with r1+r2 and |r1−r2|; equal centers require separate treatment.
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.
PT2=OP2−r2PT^2=OP^2-r^2
Worked solution

  1. Translate the geometric condition into a coordinate equation.

  2. Apply the stated relation and retain its conditions.

    PT=81−4PT=\sqrt{81-4}

  3. The external point, tangent point and center form a right triangle.

The requested value is 8.77496438739.

Checks and common pitfalls: The external point, tangent point and center form a right triangle.

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

07 / Foundation#Your turn

Classify the relation of these two circles.

C1:x2+y2=64,C2:(x−11)2+y2=9C_1:x^2+y^2=64, C_2:(x-11)^2+y^2=9
  • Compare center distance with r1+r2 and |r1−r2|; equal centers require separate treatment.
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.
d=r1+r2d=r1+r2
Worked solution

  1. Translate the geometric condition into a coordinate equation.

  2. Apply the stated relation and retain its conditions.

    d=11=r1+r2d=11=r_1+r_2

  3. The circles are externally tangent.

The requested relation or conclusion is shown below.

d=r1+r2=11d=r_1+r_2=11

Checks and common pitfalls: The circles are externally tangent.

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.

08 / Standard#Your turn

Classify the relation of these two circles.

C1:x2+y2=81,C2:(x−12)2+y2=9C_1:x^2+y^2=81, C_2:(x-12)^2+y^2=9
  • Compare center distance with r1+r2 and |r1−r2|; equal centers require separate treatment.
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.
d=r1+r2d=r1+r2
Worked solution

  1. Translate the geometric condition into a coordinate equation.

  2. Apply the stated relation and retain its conditions.

    d=12=r1+r2d=12=r_1+r_2

  3. The circles are externally tangent.

The requested relation or conclusion is shown below.

d=r1+r2=12d=r_1+r_2=12

Checks and common pitfalls: The circles are externally tangent.

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.

09 / Standard#Your turn

Find the radical axis and state when it is a common chord line.

x2+y2=25,(x−10)2+y2=25x^2+y^2=25,\quad(x-10)^2+y^2=25
  • Compare center distance with r1+r2 and |r1−r2|; equal centers require separate treatment.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Translate the geometric condition into a coordinate equation.
Hint 2
Subtract the equations.
Worked solution

  1. Translate the geometric condition into a coordinate equation.

  2. Apply the stated relation and retain its conditions.

    −2(10)x+100=0-2(10)x+100=0

  3. Apply the stated relation and retain its conditions.

    x=10/2x=10/2

  4. The center distance is 10. The circles are tangent, so there is no nonzero common chord.

The requested relation or conclusion is shown below.

x=5x=5

Checks and common pitfalls: The center distance is 10. The circles are tangent, so there is no nonzero common chord.

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 tangent at the given point.

x2+y2=3025,P=(33,44)x^2+y^2=3025, P=(33,44)
  • Compare center distance with r1+r2 and |r1−r2|; equal centers require separate treatment.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Translate the geometric condition into a coordinate equation.
Hint 2
The radius is normal to the tangent.
Worked solution

  1. Translate the geometric condition into a coordinate equation.

  2. Apply the stated relation and retain its conditions.

    3(x−33)+4(y−44)=03(x-33)+4(y-44)=0

  3. Apply the stated relation and retain its conditions.

    3x+4y=2753x+4y=275

  4. The point is on the circle and fixes the tangent constant.

The requested relation or conclusion is shown below.

3x+4y=2753x+4y=275

Checks and common pitfalls: The point is on the circle and fixes the tangent constant.

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

Find all values of a for which y=a is tangent to x²+y²=t².

t=12t=12
  • Compare center distance with r1+r2 and |r1−r2|; equal centers require separate treatment.
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.
d=∣a∣=rd=|a|=r
Worked solution

  1. Translate the geometric condition into a coordinate equation.

  2. Apply the stated relation and retain its conditions.

    ∣a∣=12⇒a=±12|a|=12\Rightarrow a=\pm12

  3. Both the upper and lower horizontal tangents must be retained.

The requested relation or conclusion is shown below.

a=±12a=\pm12

Checks and common pitfalls: Both the upper and lower horizontal tangents must be retained.

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 tangent at the given point.

x2+y2=4225,P=(39,52)x^2+y^2=4225, P=(39,52)
  • Compare center distance with r1+r2 and |r1−r2|; equal centers require separate treatment.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Translate the geometric condition into a coordinate equation.
Hint 2
The radius is normal to the tangent.
Worked solution

  1. Translate the geometric condition into a coordinate equation.

  2. Apply the stated relation and retain its conditions.

    3(x−39)+4(y−52)=03(x-39)+4(y-52)=0

  3. Apply the stated relation and retain its conditions.

    3x+4y=3253x+4y=325

  4. The point is on the circle and fixes the tangent constant.

The requested relation or conclusion is shown below.

3x+4y=3253x+4y=325

Checks and common pitfalls: The point is on the circle and fixes the tangent constant.

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

Find all values of a for which y=a is tangent to x²+y²=t².

t=14t=14
  • Compare center distance with r1+r2 and |r1−r2|; equal centers require separate treatment.
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.
d=∣a∣=rd=|a|=r
Worked solution

  1. Translate the geometric condition into a coordinate equation.

  2. Apply the stated relation and retain its conditions.

    ∣a∣=14⇒a=±14|a|=14\Rightarrow a=\pm14

  3. Both the upper and lower horizontal tangents must be retained.

The requested relation or conclusion is shown below.

a=±14a=\pm14

Checks and common pitfalls: Both the upper and lower horizontal tangents must be retained.

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

    • Classify intersection and tangency using distances, discriminants and common chords.
    • Which condition is essential in relations between lines and circles?
    • How many intersection points remain when a secant is moved away from the circle?

    Board plan

    • Line-circle relation: Compare center-to-line distance with radius; a secant chord has half-length √(r²−d²).
    • Circle-circle conditions: Compare center distance with r1+r2 and |r1−r2|; equal centers require separate treatment.

    Anticipated thinking

    • Equal center distance and radius difference is internal tangency only for distinct centers and positive radii.

    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 ↗