LEARN · EXPLAIN · REVISE

Find the tangent length from P to the circle.

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 / Foundation#Your turn

P=(9,0),\quad x^2+y^2=4

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

Use this intermediate relation.

PT^2=OP^2-r^2

Worked solution

The requested value is 8.77496438739.

Checks and common pitfalls: The external point, tangent point and center form a right triangle.

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.