LEARN · EXPLAIN · REVISE

Find the largest value of 3x+4y on the ellipse.

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

高二選擇性必修第一册(A版).pdf · 3.1 · PDF 110 / printed page 105

TOPIC 01

Ellipse

Use the focal definition, standard equation and geometric properties of an ellipse.

Use the focal definition, standard equation and geometric properties of an ellipse.
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#Worked example

x^2/36+y^2/16=1

For a horizontal major axis a>b>0; the larger denominator specifies the major axis.

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.

x=a cos u, y=b sin u

Worked solution

The requested value is 24.0831891576.

Checks and common pitfalls: The full trigonometric range attains the amplitude.

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

Teacher preparation and assessment

Use the focal definition, standard equation and geometric properties of an ellipse.
Which condition is essential in ellipse?
Predict the ellipse shape and eccentricity as b approaches a or zero.

Focal sum: Distances to the two foci sum to 2a; c²=a²−b².
$x^2/a^2+y^2/b^2=1$
Axis conditions: For a horizontal major axis a>b>0; the larger denominator specifies the major axis.