LEARN · EXPLAIN · REVISE

Find the fixed point shared by this line family.

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

TOPIC 01

Equations of a line

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

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

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

01 / Transfer#Worked example

(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

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

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

Teacher preparation and assessment

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?

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.