LEARN · EXPLAIN · REVISE

The line y=x+1 meets x²−y²=1 once. Is it tangent? Justify.

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

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

Revisit first: Ellipse: definition, axes and constraints

TOPIC 01

Hyperbola: branches and asymptotes

Keep signed and absolute focal differences separate; inspect degree loss when a line is parallel to an asymptote.

What you will be able to explain

Derive c²=a²+b² from the focal difference.
Identify branches, vertices and asymptotes in both orientations.
Classify focal-difference boundary cases and one-intersection lines.

Try it. Leave your reasoning visible.

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

Examples 1–3: study a full demonstration; finish a solution from its first step; then choose a method independently before opening the solution. The remaining exercises may continue after class.

01 / Transfer#Worked example

The line y=x+1 meets x²−y²=1 once. Is it tangent? Justify.

Skills and prerequisite lessons

Scope: JM01 / JM02

Working and explanation

BUILD THE REASONING

Hint 1

Substitution may reduce the degree.

Hint 2

Compare the actual tangent slope at the intersection.

Worked solution

The quadratic terms cancel, leaving one root.
$x^2-(x+1)^2=1\Rightarrow x=-1,\quad y=0$
The tangent at (−1,0) is vertical, whereas y=x+1 is not.
$x=-1$

Not tangent; it is parallel to an asymptote.

Checks and common pitfalls: One intersection does not imply tangency when the substituted equation becomes linear.

Reasoning checklist · self / teacher assessment

Not tangent; it is parallel to an asymptote.
One intersection does not imply tangency when the substituted equation becomes linear.

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

Teacher preparation and assessment

40-minute lesson

Minutes	Focus	Teaching and student action
5	Prediction and diagnosis	Locate the sign of the focal difference at each vertex.
8	Concept derivation	Derive and sketch asymptote slopes with horizontal and vertical axes.
10	Guided application	Calculate a point and test whether an asymptote intersects.
12	Transfer and explanation	Classify the offset-line and focal-difference boundary cases.
5	Exit evidence	Explain why a single intersection is insufficient for tangency.

Question sequence

Which focus is farther on the right branch?
Can a line have one intersection without being tangent?

Board development

Tabulate absolute difference 0, between 0 and 2c, equal to 2c and greater than 2c.
Substitute y=x+k before mentioning any discriminant.

Expected correct and incorrect responses

Students set the discriminant to zero even after the x² term disappears. Circle the actual leading coefficient first.
Correct: “The quadratic coefficient cancels, so one intersection need not be tangency.”
Incorrect: “A line with exactly one intersection must be tangent.”

Follow-up questions

How does the unique intersection move as the parallel line offset tends to zero?

Remediation

Use the vertices to decide which signed focal difference is positive.

Exit check

Give an equation of a one-intersection non-tangent line and explain the degree loss.

Assessment checklist and sample responses

Require branch sign, standard-form parameters and a separate tangency justification. Sample: substituting y=x+1 gives −2x−1=1, so x=−1; this linear equation alone does not imply tangency.
Correct: “The quadratic coefficient cancels, so one intersection need not be tangency.”