LEARN · EXPLAIN · REVISE

Show there is one common point and explain why this line is not tangent.

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

高二選擇性必修第一册(A版).pdf · 3.2 · PDF 123 / printed page 118

Revisit first: Ellipse

TOPIC 01

Hyperbola

Use the distance-difference definition, asymptotes and branch restrictions.

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

01 / Transfer#Worked example

y=x+4, x^2-y^2=16

a,b>0 and e>1; a distance difference is absolute, and an asymptote is not part of the curve.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Translate the geometric condition into a coordinate equation.

Hint 2

Substitution cancels the quadratic terms.

Worked solution

Translate the geometric condition into a coordinate equation.
Apply the stated relation and retain its conditions.
$-8x-16=16 ⇒ x=-4, y=0$
Apply the stated relation and retain its conditions.
$\text{tangent at }P:x=-4$
The line is parallel to an asymptote; one intersection can result from degree reduction.

The requested relation or conclusion is shown below.

P=(-4,0)

Checks and common pitfalls: The line is parallel to an asymptote; one intersection can result from degree reduction.

Reasoning checklist · self / teacher assessment

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

Teacher preparation and assessment

Parameters: For x²/a²−y²/b²=1, c²=a²+b² and asymptotes are y=±(b/a)x.
Branch conditions: a,b>0 and e>1; a distance difference is absolute, and an asymptote is not part of the curve.