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A point on x=t cos s,y=2sin s maximises x+ty. Find the maximum.

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

T04高二理組數學思維本(2026).pdf · V. Parametric equations (PDF27–29) · PDF 27 / printed page 26

Revisit first: Parabola

TOPIC 01

Parametric curves and preserved domains

Eliminate parameters while retaining ranges, missing points and the geometry of the locus.

Eliminate parameters while retaining ranges, missing points and the geometry of the locus.
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

t=3

An eliminated equation may contain extra points; retain every restriction implied by the parameter domain.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Write the defining equation, retain exclusions, and then solve or compare.

Hint 2

Use this intermediate relation.

x+ty=t(cos s+2sin s)

Worked solution

Write the defining equation, retain exclusions, and then solve or compare.
Apply the stated relation and retain its conditions.
$\max=t\sqrt{1^2+2^2}=3\sqrt5$
The full angle range permits the amplitude maximum.

The requested value is 6.7082039325.

Checks and common pitfalls: The full angle range permits the amplitude maximum.

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

Teacher preparation and assessment

Eliminate parameters while retaining ranges, missing points and the geometry of the locus.
Which condition is essential in parametric curves and preserved domains?
Do x=s²,y=s⁴ and x=s,y=s² trace the same set?

Model or definition: Eliminate parameters while retaining ranges, missing points and the geometry of the locus.
$x=f(s),\quad y=g(s)$
Conditions: An eliminated equation may contain extra points; retain every restriction implied by the parameter domain.