Kenneth ChengMathematical modeling
← Senior Mathematics Studio

LEARN · EXPLAIN · REVISE

Equations of a circle

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

高二選擇性必修 第一册(A版).pdf · 2.4 · PDF 87 / printed page 82

Revisit first: Intersections and distance formulas

TOPIC 01

Equations of a circle

Construct circle equations from centers, radii, diameters and point constraints.

What you will be able to explain

  • Construct circle equations from centers, radii, diameters and point constraints.
  • Justify the method and check the conditions in a new situation.

Standard form

A circle is the locus at fixed positive distance from its center.

(x−h)2+(y−k)2=r2(x-h)^2+(y-k)^2=r^2

Existence

In x²+y²+Dx+Ey+F=0, a nondegenerate circle requires (D²+E²)/4−F>0.

PREDICT → EXPLORE → EXPLAIN → TRANSFER

Make a prediction, then explore the relationship.

Lesson question: Predict the shape when the constant of a circle equation changes through the zero-radius threshold.

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: Compare a circle, a point and an empty real locus by completing squares.

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 radius of the circle.

(x−2)2+(y+1)2=4(x-2)^2+(y+1)^2=4
  • In x²+y²+Dx+Ey+F=0, a nondegenerate circle requires (D²+E²)/4−F>0.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Complete squares before reading the geometric parameters.
Hint 2
Use this intermediate relation.
r2=t2r^2=t^2
Worked solution

  1. Complete squares before reading the geometric parameters.

  2. Apply the stated relation and retain its conditions.

    r=4=2r=\sqrt{4}=2

  3. The radius is the positive square root.

The requested value is 2.

Checks and common pitfalls: The radius is the positive square root.

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

02 / Standard#Worked example

Find the parameter range for a real nondegenerate circle.

x2+y2−6x+4y+λ=0x^2+y^2-6x+4y+\lambda=0
  • In x²+y²+Dx+Ey+F=0, a nondegenerate circle requires (D²+E²)/4−F>0.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Complete squares before reading the geometric parameters.
Hint 2
Use this intermediate relation.
r2=t2+4−λr²=t²+4−λ
Worked solution

  1. Complete squares before reading the geometric parameters.

  2. Apply the stated relation and retain its conditions.

    (x−3)2+(y+2)2=13−λ(x-3)^2+(y+2)^2=13-\lambda

  3. Apply the stated relation and retain its conditions.

    r2>0r^2>0

  4. The zero-radius boundary is a point rather than a circle.

The requested relation or conclusion is shown below.

λ<13\lambda<13

Checks and common pitfalls: The zero-radius boundary is a point rather than a circle.

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

The circle center is (t,0), radius 2. Find the largest x-coordinate on it.

t=4t=4
  • In x²+y²+Dx+Ey+F=0, a nondegenerate circle requires (D²+E²)/4−F>0.
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−t∣≤2|x-t|\le2
Worked solution

  1. Translate the geometric condition into a coordinate equation.

  2. Apply the stated relation and retain its conditions.

    xmax⁡=4+2=6x_{\max}=4+2=6

  3. The extremal point lies on the horizontal diameter.

The requested value is 6.

Checks and common pitfalls: The extremal point lies on the horizontal diameter.

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

04 / Foundation#Your turn

Find the radius of the circle.

(x−2)2+(y+1)2=25(x-2)^2+(y+1)^2=25
  • In x²+y²+Dx+Ey+F=0, a nondegenerate circle requires (D²+E²)/4−F>0.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Complete squares before reading the geometric parameters.
Hint 2
Use this intermediate relation.
r2=t2r^2=t^2
Worked solution

  1. Complete squares before reading the geometric parameters.

  2. Apply the stated relation and retain its conditions.

    r=25=5r=\sqrt{25}=5

  3. The radius is the positive square root.

The requested value is 5.

Checks and common pitfalls: The radius is the positive square root.

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

05 / Foundation#Your turn

Find the x-coordinate of the center.

x2+y2−12x+4y+1=0x^2+y^2-12x+4y+1=0
  • In x²+y²+Dx+Ey+F=0, a nondegenerate circle requires (D²+E²)/4−F>0.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Complete squares before reading the geometric parameters.
Hint 2
Use this intermediate relation.
x2−2tx=(x−t)2−t2x^2-2tx=(x-t)^2-t^2
Worked solution

  1. Complete squares before reading the geometric parameters.

  2. Apply the stated relation and retain its conditions.

    (x−6)2+(y+2)2=39(x-6)^2+(y+2)^2=39

  3. The center has the opposite signs from the offsets inside the squares.

The requested value is 6.

Checks and common pitfalls: The center has the opposite signs from the offsets inside the squares.

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

06 / Foundation#Your turn

Find the squared radius of the circle with diameter endpoints A and B.

A=(0,0),B=(14,4)A=(0,0),\quad B=(14,4)
  • In x²+y²+Dx+Ey+F=0, a nondegenerate circle requires (D²+E²)/4−F>0.
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.
r2=AB2/4r^2=AB^2/4
Worked solution

  1. Translate the geometric condition into a coordinate equation.

  2. Apply the stated relation and retain its conditions.

    AB2=(2⋅7)2+42AB^2=(2\cdot7)^2+4^2

  3. Apply the stated relation and retain its conditions.

    r2=53r^2=53

  4. The radius is half the diameter, so its square is one quarter.

The requested value is 53.

Checks and common pitfalls: The radius is half the diameter, so its square is one quarter.

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

07 / Foundation#Your turn

Find the circle centered at (t,−1) and passing through (t+3,3).

t=8t=8
  • In x²+y²+Dx+Ey+F=0, a nondegenerate circle requires (D²+E²)/4−F>0.
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.
r2=32+42r^2=3^2+4^2
Worked solution

  1. Translate the geometric condition into a coordinate equation.

  2. Apply the stated relation and retain its conditions.

    r2=25r^2=25

  3. Use the center-to-point distance; the point itself is not the center.

The requested relation or conclusion is shown below.

(x−8)2+(y+1)2=25(x-8)^2+(y+1)^2=25

Checks and common pitfalls: Use the center-to-point distance; the point itself is not the center.

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

Find the circle centered at (t,−1) and passing through (t+3,3).

t=9t=9
  • In x²+y²+Dx+Ey+F=0, a nondegenerate circle requires (D²+E²)/4−F>0.
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.
r2=32+42r^2=3^2+4^2
Worked solution

  1. Translate the geometric condition into a coordinate equation.

  2. Apply the stated relation and retain its conditions.

    r2=25r^2=25

  3. Use the center-to-point distance; the point itself is not the center.

The requested relation or conclusion is shown below.

(x−9)2+(y+1)2=25(x-9)^2+(y+1)^2=25

Checks and common pitfalls: Use the center-to-point distance; the point itself is not the center.

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 parameter range for a real nondegenerate circle.

x2+y2−20x+4y+λ=0x^2+y^2-20x+4y+\lambda=0
  • In x²+y²+Dx+Ey+F=0, a nondegenerate circle requires (D²+E²)/4−F>0.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Complete squares before reading the geometric parameters.
Hint 2
Use this intermediate relation.
r2=t2+4−λr²=t²+4−λ
Worked solution

  1. Complete squares before reading the geometric parameters.

  2. Apply the stated relation and retain its conditions.

    (x−10)2+(y+2)2=104−λ(x-10)^2+(y+2)^2=104-\lambda

  3. Apply the stated relation and retain its conditions.

    r2>0r^2>0

  4. The zero-radius boundary is a point rather than a circle.

The requested relation or conclusion is shown below.

λ<104\lambda<104

Checks and common pitfalls: The zero-radius boundary is a point rather than a circle.

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 circle through O=(0,0), A=(2t,0), B=(0,2).

t=11t=11
  • In x²+y²+Dx+Ey+F=0, a nondegenerate circle requires (D²+E²)/4−F>0.
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.
F=0;D=−2t;E=−2F=0;\quad D=-2t;\quad E=-2
Worked solution

  1. Translate the geometric condition into a coordinate equation.

  2. Apply the stated relation and retain its conditions.

    484+22D=0⇒D=−22484+22D=0\Rightarrow D=-22

  3. Apply the stated relation and retain its conditions.

    4+2E=0⇒E=−24+2E=0\Rightarrow E=-2

  4. Three noncollinear points determine one circle.

The requested relation or conclusion is shown below.

x2+y2−22x−2y=0x^2+y^2-22x-2y=0

Checks and common pitfalls: Three noncollinear points determine one circle.

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

The circle center is (t,0), radius 2. Find the largest x-coordinate on it.

t=12t=12
  • In x²+y²+Dx+Ey+F=0, a nondegenerate circle requires (D²+E²)/4−F>0.
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−t∣≤2|x-t|\le2
Worked solution

  1. Translate the geometric condition into a coordinate equation.

  2. Apply the stated relation and retain its conditions.

    xmax⁡=12+2=14x_{\max}=12+2=14

  3. The extremal point lies on the horizontal diameter.

The requested value is 14.

Checks and common pitfalls: The extremal point lies on the horizontal diameter.

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

12 / Transfer#Your turn

Find the circle through O=(0,0), A=(2t,0), B=(0,2).

t=13t=13
  • In x²+y²+Dx+Ey+F=0, a nondegenerate circle requires (D²+E²)/4−F>0.
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.
F=0;D=−2t;E=−2F=0;\quad D=-2t;\quad E=-2
Worked solution

  1. Translate the geometric condition into a coordinate equation.

  2. Apply the stated relation and retain its conditions.

    676+26D=0⇒D=−26676+26D=0\Rightarrow D=-26

  3. Apply the stated relation and retain its conditions.

    4+2E=0⇒E=−24+2E=0\Rightarrow E=-2

  4. Three noncollinear points determine one circle.

The requested relation or conclusion is shown below.

x2+y2−26x−2y=0x^2+y^2-26x-2y=0

Checks and common pitfalls: Three noncollinear points determine one circle.

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

The circle center is (t,0), radius 2. Find the largest x-coordinate on it.

t=14t=14
  • In x²+y²+Dx+Ey+F=0, a nondegenerate circle requires (D²+E²)/4−F>0.
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−t∣≤2|x-t|\le2
Worked solution

  1. Translate the geometric condition into a coordinate equation.

  2. Apply the stated relation and retain its conditions.

    xmax⁡=14+2=16x_{\max}=14+2=16

  3. The extremal point lies on the horizontal diameter.

The requested value is 16.

Checks and common pitfalls: The extremal point lies on the horizontal diameter.

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

Focus on one question

End-of-lesson check and correction

Review your latest checked answers and explanations. A draft change requires a fresh check. Written work needs your self-assessment or a teacher’s review.

Enable JavaScript for a summary of your local work.

    Choose a foundation skill to revisit ↗

    Teacher preparation and assessment

    Question sequence

    • Construct circle equations from centers, radii, diameters and point constraints.
    • Which condition is essential in equations of a circle?
    • Predict the shape when the constant of a circle equation changes through the zero-radius threshold.

    Board plan

    • Standard form: A circle is the locus at fixed positive distance from its center.
      (x−h)2+(y−k)2=r2(x-h)^2+(y-k)^2=r^2
    • Existence: In x²+y²+Dx+Ey+F=0, a nondegenerate circle requires (D²+E²)/4−F>0.

    Anticipated thinking

    • A positive squared radius is necessary; zero gives a single point, and negative gives no real locus.

    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.

    No sign-in. Work stays in this browser. Export before clearing browser data. Written reasoning is assessed with a checklist.

    Curriculum and source notes ↗