LEARN · EXPLAIN · REVISE

Find the tangent of slope 1 to y²=4tx.

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

高二選擇性必修第一册(A版).pdf · 3.3 · PDF 135 / printed page 130

Revisit first: Hyperbola

TOPIC 01

Parabola

Relate focus, directrix and vertex, and retain conditions in line-parabola intersections.

Relate focus, directrix and vertex, and retain conditions in line-parabola intersections.
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 / Transfer#Your turn

t=13

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.

y=x+b\Rightarrow x^2+(2b-4t)x+b^2=0

Worked solution

Translate the geometric condition into a coordinate equation.
Apply the stated relation and retain its conditions.
$\Delta=(2b-52)^2-4b^2=208(13-b)$
Apply the stated relation and retain its conditions.
$\Delta=0\Rightarrow b=13$
The quadratic coefficient remains nonzero, so a repeated root establishes tangency.

The requested relation or conclusion is shown below.

y=x+13

Checks and common pitfalls: The quadratic coefficient remains nonzero, so a repeated root establishes tangency.

Reasoning checklist · self / teacher assessment

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

Teacher preparation and assessment

Relate focus, directrix and vertex, and retain conditions in line-parabola intersections.
Which condition is essential in parabola?
How does changing the sign of a alter y²=4ax?

Focus-directrix: y²=4ax has focus (a,0) and directrix x=−a.
Nondegeneracy: a must be nonzero; a parabola has a vertex and no center of symmetry.