LEARN · EXPLAIN · REVISE

Equations of a line

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

高二選擇性必修第一册(A版).pdf · 2.2 · PDF 64 / printed page 59

Revisit first: Inclination and slope of a line

TOPIC 01

Equations of a line

Choose point-slope, two-point, intercept or general form without losing exceptional lines.

What you will be able to explain

Choose point-slope, two-point, intercept or general form without losing exceptional lines.
Justify the method and check the conditions in a new situation.

General form

Ax+By+C=0 represents a line when A and B are not both zero.

A(x-x_0)+B(y-y_0)=0

Form restrictions

Point-slope form omits vertical lines; intercept form requires nonzero intercepts.

PREDICT → EXPLORE → EXPLAIN → TRANSFER

Make a prediction, then explore the relationship.

Lesson question: Which common forms can describe a line through the origin or a vertical line?

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

y=1x+1. Angle is measured modulo 180°; distance from the origin=0.7071. This slope form excludes vertical lines.

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

Transfer: Compare equivalent forms and identify where division excludes a case.

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 line through (2,3) with slope t.

t=2

Point-slope form omits vertical lines; intercept form requires nonzero intercepts.

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.

y-y_0=m(x-x_0)

Worked solution

Translate the geometric condition into a coordinate equation.
Apply the stated relation and retain its conditions.
$y-3=2(x-2)$
Substitution of the given point checks the constant.

The requested relation or conclusion is shown below.

y-3=2(x-2)

Checks and common pitfalls: Substitution of the given point checks the 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.

02 / Standard#Worked example

Find the perpendicular bisector of (0,0) and (2t,0).

t=3

Point-slope form omits vertical lines; intercept form requires nonzero intercepts.

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.

M=(t,0),\quad AB\text{ horizontal}

Worked solution

Translate the geometric condition into a coordinate equation.
Apply the stated relation and retain its conditions.
$M=(3,0)$
Apply the stated relation and retain its conditions.
$x=3$
The perpendicular bisector consists of points equidistant from the two endpoints.

The requested relation or conclusion is shown below.

x=3

Checks and common pitfalls: The perpendicular bisector consists of points equidistant from the two endpoints.

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 fixed point shared by this line family.

(x-4)+\lambda(y-8)=0

Point-slope form omits vertical lines; intercept form requires nonzero intercepts.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Translate the geometric condition into a coordinate equation.

Hint 2

Both the constant part and parameter coefficient vanish.

Worked solution

Translate the geometric condition into a coordinate equation.
Apply the stated relation and retain its conditions.
$x=4, y=8$
The point works for every real parameter.

The requested relation or conclusion is shown below.

P=(4,8)

Checks and common pitfalls: The point works for every real parameter.

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 line through (2,3) with slope t.

t=5

Point-slope form omits vertical lines; intercept form requires nonzero intercepts.

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.

y-y_0=m(x-x_0)

Worked solution

Translate the geometric condition into a coordinate equation.
Apply the stated relation and retain its conditions.
$y-3=5(x-2)$
Substitution of the given point checks the constant.

The requested relation or conclusion is shown below.

y-3=5(x-2)

Checks and common pitfalls: Substitution of the given point checks the 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.

05 / Foundation#Your turn

Find the line through (t,1) and (t,5).

t=6

Point-slope form omits vertical lines; intercept form requires nonzero intercepts.

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_1=x_2

Worked solution

Translate the geometric condition into a coordinate equation.
Apply the stated relation and retain its conditions.
$x=6$
Use a vertical equation directly instead of dividing by zero in a slope formula.

The requested relation or conclusion is shown below.

x=6

Checks and common pitfalls: Use a vertical equation directly instead of dividing by zero in a slope formula.

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.

06 / Foundation#Your turn

Find the y-intercept of the line with x-intercept t and passing through (0,4).

t=7

Point-slope form omits vertical lines; intercept form requires nonzero intercepts.

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+y/b=1

Worked solution

Translate the geometric condition into a coordinate equation.
Apply the stated relation and retain its conditions.
$(0,4)\Rightarrow4/b=1\Rightarrow b=4$
An intercept is a signed coordinate, not a distance.

The requested value is 4.

Checks and common pitfalls: An intercept is a signed coordinate, not a distance.

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

07 / Foundation#Your turn

Find C so that Ax+By+C=0 passes through P.

A=2,\ B=-3,\ P=(8,1)

Point-slope form omits vertical lines; intercept form requires nonzero intercepts.

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.

C=-Ax_P-By_P

Worked solution

Translate the geometric condition into a coordinate equation.
Apply the stated relation and retain its conditions.
$C=-2(8)+3=-13$
A point satisfies the equation exactly when the substituted expression is zero.

The requested value is -13.

Checks and common pitfalls: A point satisfies the equation exactly when the substituted expression is zero.

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

08 / Standard#Your turn

Find C so that Ax+By+C=0 passes through P.

A=2,\ B=-3,\ P=(9,1)

Point-slope form omits vertical lines; intercept form requires nonzero intercepts.

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.

C=-Ax_P-By_P

Worked solution

Translate the geometric condition into a coordinate equation.
Apply the stated relation and retain its conditions.
$C=-2(9)+3=-15$
A point satisfies the equation exactly when the substituted expression is zero.

The requested value is -15.

Checks and common pitfalls: A point satisfies the equation exactly when the substituted expression is zero.

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

09 / Standard#Your turn

Find the perpendicular bisector of (0,0) and (2t,0).

t=10

Point-slope form omits vertical lines; intercept form requires nonzero intercepts.

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.

M=(t,0),\quad AB\text{ horizontal}

Worked solution

Translate the geometric condition into a coordinate equation.
Apply the stated relation and retain its conditions.
$M=(10,0)$
Apply the stated relation and retain its conditions.
$x=10$
The perpendicular bisector consists of points equidistant from the two endpoints.

The requested relation or conclusion is shown below.

x=10

Checks and common pitfalls: The perpendicular bisector consists of points equidistant from the two endpoints.

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 reflection of the line in the y-axis.

y=11x+2

Point-slope form omits vertical lines; intercept form requires nonzero intercepts.

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\mapsto-x

Worked solution

Translate the geometric condition into a coordinate equation.
Apply the stated relation and retain its conditions.
$y=11(-x)+2=-11x+2$
Reflection changes the x-coordinate sign while leaving y unchanged.

The requested relation or conclusion is shown below.

y=-11x+2

Checks and common pitfalls: Reflection changes the x-coordinate sign while leaving y unchanged.

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 the fixed point shared by this line family.

(x-12)+\lambda(y-24)=0

Point-slope form omits vertical lines; intercept form requires nonzero intercepts.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Translate the geometric condition into a coordinate equation.

Hint 2

Both the constant part and parameter coefficient vanish.

Worked solution

Translate the geometric condition into a coordinate equation.
Apply the stated relation and retain its conditions.
$x=12, y=24$
The point works for every real parameter.

The requested relation or conclusion is shown below.

P=(12,24)

Checks and common pitfalls: The point works for every real parameter.

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 reflection of the line in the y-axis.

y=13x+2

Point-slope form omits vertical lines; intercept form requires nonzero intercepts.

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\mapsto-x

Worked solution

Translate the geometric condition into a coordinate equation.
Apply the stated relation and retain its conditions.
$y=13(-x)+2=-13x+2$
Reflection changes the x-coordinate sign while leaving y unchanged.

The requested relation or conclusion is shown below.

y=-13x+2

Checks and common pitfalls: Reflection changes the x-coordinate sign while leaving y unchanged.

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 the fixed point shared by this line family.

(x-14)+\lambda(y-28)=0

Point-slope form omits vertical lines; intercept form requires nonzero intercepts.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Translate the geometric condition into a coordinate equation.

Hint 2

Both the constant part and parameter coefficient vanish.

Worked solution

Translate the geometric condition into a coordinate equation.
Apply the stated relation and retain its conditions.
$x=14, y=28$
The point works for every real parameter.

The requested relation or conclusion is shown below.

P=(14,28)

Checks and common pitfalls: The point works for every real parameter.

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.

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.

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Choose a foundation skill to revisit ↗

Teacher preparation and assessment

Question sequence

Choose point-slope, two-point, intercept or general form without losing exceptional lines.
Which condition is essential in equations of a line?
Which common forms can describe a line through the origin or a vertical line?

Board plan

General form: Ax+By+C=0 represents a line when A and B are not both zero.
$A(x-x_0)+B(y-y_0)=0$
Form restrictions: Point-slope form omits vertical lines; intercept form requires nonzero intercepts.

Anticipated thinking

Dividing an equation by a parameter can silently delete a valid line.

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.