LEARN · EXPLAIN · REVISE

For which k is the locus a noncircular, nondegenerate 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: definition, axes and constraints

Build an ellipse from a distance sum, identify its major direction and distinguish a genuine ellipse from boundary loci.

What you will be able to explain

Relate the focal definition to a²=b²+c².
Read translated and vertical ellipses without confusing denominators with lengths.
Use scaling to optimize a linear quantity and test locus feasibility.

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 / Standard#Your turn

For which k is the locus a noncircular, nondegenerate ellipse?

x^2/9+y^2/k=1

Skills and prerequisite lessons

Scope: JM01 / JM02

Working and explanation

BUILD THE REASONING

Hint 1

Both denominators must be positive.

Hint 2

Equal denominators give a circle.

Worked solution

Exclude zero and negative k.
$k>0$
Exclude the circle while permitting either major-axis direction.
$k\ne9$

k>0 and k≠9.

Checks and common pitfalls: k>9 is also an ellipse; it simply has a vertical major axis.

Reasoning checklist · self / teacher assessment

k>0 and k≠9.
k>9 is also an ellipse; it simply has a vertical major axis.

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	Predict the string-and-pins boundary cases.
8	Concept derivation	Derive the relation between a,b,c from axis vertices.
10	Guided application	Read horizontal, vertical and translated equations.
12	Transfer and explanation	Solve a scaled optimization and an impossible-locus test.
5	Exit evidence	Classify the focal-sum exit cases.

Question sequence

What happens when the string length equals the distance between the pins?
Which direction has the largest intercept?

Board development

Use three rows: sum below, equal to, and above the focal separation.
Scale x=2U,y=3V before bounding 3x+4y.

Expected correct and incorrect responses

Students always put foci on the x-axis. Ask them to mark the larger intercept first.
Correct: “The larger denominator is under y², so the foci lie on the y-axis.”
Incorrect: “The denominator 25 means the semiaxis has length 25.”

Follow-up questions

Why must an equality case be checked after finding an upper bound?

Remediation

Draw semiaxes as actual lengths before calculating c².

Exit check

Classify distance sums 5, 6 and 10 for foci at (±3,0).

Assessment checklist and sample responses

Credit axis identification, positive-length conditions and an attainable equality case. Sample: 6U+12V≤√180 with U²+V²=1; equality fixes the maximizing point.
Correct: “The larger denominator is under y², so the foci lie on the y-axis.”