LEARN · EXPLAIN · REVISE

A=(0,1), B=(t,3), P is on y=0. Find the minimum of AP+PB.

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

高二選擇性必修第一册(A版).pdf · 2.3 · PDF 75 / printed page 70

TOPIC 01

Intersections and distance formulas

Solve intersections and calculate point-line and parallel-line distances with geometric checks.

Solve intersections and calculate point-line and parallel-line distances with geometric checks.
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 / Standard#Your turn

t=12

The denominator is the norm of the normal vector; parallel-line formulas require matched coefficients.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Translate the geometric condition into a coordinate equation.

Hint 2

Reflect A to A\prime=(0,-1).

Worked solution

Translate the geometric condition into a coordinate equation.
Apply the stated relation and retain its conditions.
$AP=A'P$
Apply the stated relation and retain its conditions.
$AP+PB\ge A'B=\sqrt{12^2+4^2}$
The segment A′B crosses y=0, so equality is attainable.

The requested value is 12.6491106407.

Checks and common pitfalls: The segment A′B crosses y=0, so equality is attainable.

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

Teacher preparation and assessment

Solve intersections and calculate point-line and parallel-line distances with geometric checks.
Which condition is essential in intersections and distance formulas?
Why does multiplying a line equation by 100 leave its distance formula unchanged?

Intersection: Solve both linear equations simultaneously and check both.
Normalized distance: The denominator is the norm of the normal vector; parallel-line formulas require matched coefficients.
$d=|Ax_0+By_0+C|/\sqrt{A^2+B^2}$

Subtracting constants before matching parallel-line coefficients gives a wrong distance.