LEARN · EXPLAIN · REVISE

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

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

TOPIC 01

Relations between lines and circles

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

Classify intersection and tangency using distances, discriminants and common chords.
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#Worked example

x^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

Translate the geometric condition into a coordinate equation.
Apply the stated relation and retain its conditions.
$-2(3)x+9=0$
Apply the stated relation and retain its conditions.
$x=3/2$
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=1.5

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

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

Teacher preparation and assessment

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?

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.

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