Cuts through the vertex, where the curve degenerates

What to assume they know

• One surface, four curves, chosen by the angle of the cut — the cone's construction, the words vertex, axis, generator and nappe, and the four angle conditions of §10.2.1
• That a plane and a line either meet at one point, or the plane contains the whole line, or they never meet — the trichotomy the whole section runs on
• Reading a strict inequality against an inclusive one, and noticing when an endpoint changes hands

What they should be able to do

• State the condition §10.2.1 imposes on the cutting plane, and say why §10.2.2 is the complementary case rather than an exception
• List the three degenerate sections and the angle condition producing each
• Explain why a plane through the vertex can contain a generator entirely, and why that is what turns a curve into a line
• Explain why the ellipse case and the circle case merge into one degenerate case, and identify which endpoint changes from strict to inclusive
• Read the two panels of Fig 10.10 and say what distinguishes them and what does not
• Identify which degenerate cases the chapter names as degenerations of a named conic and which it leaves unattached
• Distinguish the cone's degenerate parabola from the separate degenerate case the focus–directrix definition produces in §10.4

Where it usually goes wrong

• "Degenerate means the maths went wrong." Nothing went wrong. The plane was moved to a legal position and the definition returned an honest answer that happens not to be a curve. Every case in §10.2.2 is a genuine intersection of a genuine plane with a genuine cone.
• "A point is not a conic section." The definition on p. 177 calls a conic section whatever curve a plane cuts from a right circular cone, and this is exactly that. The chapter's own §10.2.2 lists it. Whether it deserves the name is a question about naming, not about geometry.
• "There should be four degenerate cases to match the four curves." There are three, and the reason is stated above: pinning the plane to the vertex destroys the distinction 90° was drawing.
• "The straight line in Fig 10.9 is a tangent to something." It is a generator — a line lying entirely inside the surface. The plane touches the cone along the whole of it and crosses to neither side.
• "Panels (a) and (b) of Fig 10.10 show different sections." They show the same kind of section under different planes. The lettering inside the artwork distinguishes β = 0 from β > 0; the answer is a crossed pair of lines in both.
• "The vertex case is a special sub-case of the ordinary cases." It is the complementary case. §10.2.1 and §10.2.2 between them exhaust where a plane can meet a cone, which is why the chapter needed both before moving on to equations.
• "§10.4's degenerate parabola is the same object as Fig 10.9's." Both are straight lines, and that is where the resemblance ends. One is a generator of a cone; the other is fixed by a focus lying on its directrix. Teaching them as one thing loses the observation that two different definitions of parabola both degenerate to a line.

Questions to check understanding

• Given the plane passes through the vertex and a value of β relative to α, name the section
• State the condition under which the section of a cone is a single point
• Explain why a vertex cut with β = α yields a single line, arguing from generators
• Explain why there are three degenerate cases and not four
• Identify, from a drawing, whether the plane passes through the vertex
• Short-answer: give two different ways a parabola can degenerate to a straight line, and say what is different about them

Examples worth working on the board

Definitional again — no numbers. The printed conditions and the figures are the content. Every inequality below was read off the page image individually.

• The three printed conditions (§10.2.2, p. 178), with the plane now passing through the vertex:
  • α < β ≤ 90° → a point (Fig 10.8). Strict at α, inclusive at 90°.
  • β = α → a single straight line, drawn in Fig 10.9, which the chapter records as the degenerate case of a parabola
  • 0 ≤ β < α → a crossed pair of straight lines, drawn in Fig 10.10 and recorded as the degenerate case of a hyperbola
• The merge, which is the topic's best single fact. In §10.2.1 the range above α was split at 90°: strictly below gave an ellipse, exactly 90° gave a circle. In §10.2.2 that split is gone and the two are written as one condition with 90° pulled inside it. Verified by comparing the two printed lists on the page images of pp. 177 and 178: the ellipse condition's upper end is strict, the point condition's upper end is inclusive, and the circle has no line of its own in §10.2.2 at all.
• Fig 10.10 is two figures, and the text layer cannot see it. The printed page shows panels (a) and (b) sharing the single caption Fig 10.10, with the distinguishing conditions lettered inside the artwork: panel (a) is marked 0 = β < α and panel (b) is marked 0 < β < α. Neither string appears in the extracted text of p. 179. Confirmed. So the chapter is making a point it never writes in prose — the plane may or may not contain the axis, and the section is a crossed pair of lines either way.
• Which labels are printed and which are not. Read off the p. 178 image: the chapter attaches "degenerated case of a parabola" to case (b) and "degenerated case of a hyperbola" to case (c). It attaches no such phrase to case (a). So neither "degenerate ellipse" nor "degenerate circle" is printed in §10.2.2.
• The second, unrelated degeneration (§10.4 Note, p. 182). If the parabola's fixed point is allowed to sit on its fixed line, the set of points equidistant from both is the line through the fixed point perpendicular to the fixed line. The chapter calls this a degenerate case of the parabola too. Verified as distinct: this argument uses only the focus and directrix and never mentions a cone, and the line it produces is perpendicular to the directrix, whereas the line in Fig 10.9 is a generator of the cone. Same word, same kind of collapsed object, two independent routes.
• Verified count: §10.2.1 lists four cases, §10.2.2 lists three. The arithmetic of the merge accounts for the missing fourth exactly.

Figures to have open

• The cone drawn with many generators, all meeting at the vertex. Not a printed figure; it is what makes sections 2 and 6 obvious rather than asserted.
• The chapter's Figs 10.8, 10.9 and 10.10 (p. 179), redrawn as schematics sharing one cone and one viewpoint. Fig 10.10 must be redrawn as two panels, with the conditions 0 = β < α and 0 < β < α carried over, because the printed distinction lives inside the artwork and is invisible in any text extraction.
• A side-by-side of the §10.2.1 condition list and the §10.2.2 condition list, with the 90° endpoint highlighted as it moves from strict to inclusive. The chapter prints the two lists a page apart and never sets them against each other; the comparison is the topic.
• A drawing of a focus sitting on its own directrix, with the perpendicular line it produces — for §10.4's separate degeneration. Standard schematic.

Where this sits in the book

• NCERT Mathematics, Textbook for Class XI, Chapter 10 "Conic Sections", §10.2.2 Degenerated conic sections (p. 178, with Figs 10.8–10.10 carried over to p. 179)
• §10.2.1 (p. 177) for the excluded-vertex clause and the four conditions this topic compares against
• Figures: Fig 10.8, Fig 10.9, Fig 10.10 panels (a) and (b), all on p. 179
• Deliberate forward reference inside the chapter: the Note in §10.4 (p. 182) gives the parabola a second, unrelated degenerate case from the focus–directrix definition; that definition is Balancing a point against a line, and the four equations that result's subject
• The closing sentence of §10.2.2 (p. 179) announces that the remaining sections will define each conic by geometric properties instead of by cutting a cone — which is why the cone does not reappear after this page