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TOPIC 01
Conic intersections, chords and inverse problems
Solve line–conic systems with degree and domain checks; use Vieta for midpoint loci and reconstruct a chord from its midpoint.
What you will be able to explain
Retain all roots and substitute them into the original system.
Use the discriminant only for an actual quadratic.
Use root sums and products without unnecessarily solving every endpoint.
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.
Checks and common pitfalls: Dividing by x would lose the vertex.
Reasoning checklist · self / teacher assessment
(0,0) and (4,4).
Dividing by x would lose the vertex.
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
Find the lost zero root in a proposed solution.
8
Concept derivation
Derive the ellipse-line quadratic and Vieta midpoint.
10
Guided application
Classify secant, tangent, external and degree-loss cases.
12
Transfer and explanation
Reconstruct a chord and prove the full midpoint locus including the vertical case.
5
Exit evidence
Write the degree condition beside the discriminant rule.
Question sequence
What is the actual degree after substitution?
Can the midpoint be found without writing both radicals?
Board development
Place degree check before discriminant, then back-substitution after roots.
For the midpoint locus, reserve a box for the vertical chord and the reverse parametrization.
Expected correct and incorrect responses
Students prove only that chord midpoints satisfy an ellipse equation. Ask how an arbitrary point on that ellipse recovers a chord.
Correct: “I keep x=0 when factoring x(x−4)=0, and then use y=x.”
Incorrect: “Dividing by x gives x=4, so (4,4) is the only intersection.”
Follow-up questions
Which datum determines the slope in the inverse-parabola chord problem?
Remediation
Factor x(x−4)=0 before any division; check each point in y=x.
Exit check
State the extra check needed before using Δ=0 as a tangency test.
Assessment checklist and sample responses
Credit degree, full root retention, back-substitution and a two-way locus argument. Sample: finite m covers k>0; the vertical chord supplies (0,0), completing the midpoint locus.
Correct: “I keep x=0 when factoring x(x−4)=0, and then use y=x.”
