One surface, four curves, chosen by the angle of the cut

What to assume they know

• What it means for a line to be inclined to another line at a fixed angle, and that rotating one line about another sweeps a surface
• That a plane and a surface can meet in a curve, and that the curve depends on how the plane is placed
• Degree measure, and comparison of angles by size — the whole classification is three inequalities and two equalities
• Steepness as the tangent of an angle, and the one line that has none — the habit of measuring a line's direction as an angle against a fixed reference, which §10.2 repeats with a cone's axis as the reference

What they should be able to do

• Describe how the double-napped right circular cone is generated, and name the role of each of the two lines involved
• Identify vertex, axis, generator and nappe on a drawing of the cone
• State which angle β measures, and against what
• Given a value of β relative to α, name which of the four sections results
• Explain why exactly two of the four cases are single values of β and two are intervals
• Explain why the section stops being a closed curve precisely at β = α
• Explain why the hyperbola's two branches come from one plane meeting one surface, not from two separate cuts
• State the condition §10.2.1 imposes on where the plane meets the cone, and why it has to be imposed

Where it usually goes wrong

• "A circle is one of the ellipses." Not in §10.2.1 as printed. The circle gets β = 90° and the ellipse gets β strictly below 90°, so on this page the two cases are disjoint and the circle is listed on its own. Whether a circle counts as an ellipse elsewhere is a different question about a different definition.
• "β is how much the plane is tilted from horizontal." It is measured against the cone's axis, which the chapter draws vertical. Horizontal is what β = 90° happens to look like in the figure, not what β means.
• "The cone is a solid." It is a hollow surface, and only the surface is cut. A solid cone would give a filled region, not a curve.
• "The cone stops at the base circle." The drawn ellipses at top and bottom of Fig 10.2 are the artist's way of ending a picture. The surface runs on indefinitely both ways, which is what lets the ellipse case stay bounded while the parabola case does not.
• "The parabola is a range of tilts." It is one exact value. Nudge β either way and the section becomes an ellipse or a hyperbola. That fragility is worth showing by sweeping β continuously.
• "Two branches means the plane was used twice." One plane, one surface, one intersection. The branches are two because the surface has two nappes and a shallow enough plane reaches both.
• "Any plane through the cone gives one of these four." Only if it misses the vertex. §10.2.1 opens by excluding the vertex, and Cuts through the vertex, where the curve degenerates is about what happens when you put it back.

Questions to check understanding

• Given α and β as numbers, name the section
• Given a named section, state the condition on β that produces it
• Identify vertex, axis, generator and nappe on an unlabelled drawing
• Explain why the parabola case is an equality and the hyperbola case an inequality
• Explain why a plane parallel to the axis gives a hyperbola and not a parabola
• Short-answer: why must the plane avoid the vertex for §10.2.1 to hold

Examples worth working on the board

This section of the chapter carries no numerical examples at all — it is definitional. The data below are the printed conditions and the figures, which are the topic's whole content.

• Generating the surface (§10.2, p. 176, Fig 10.1). One vertical line l is held fixed. A second line m crosses it at a point V, making a constant angle α with it. Now spin m about l, keeping α unchanged. Fig 10.1 shows only the two lines, the point V and the marked angle α — not the finished cone. That is deliberate: the figure is the recipe, not the result.
• The finished surface (Fig 10.2, p. 177). Labelled on the drawing itself: the axis, a generator m, the vertex V, and the upper and lower nappes. The cone is hollow and extends without limit in both directions.
• The second angle (Fig 10.3, p. 177). A plane drawn cutting the cone, with β marked between the plane and the axis. Both α and β appear in this one figure, which is the only place the two are drawn together before the four cases split them apart.
• The four printed conditions (§10.2.1, p. 177). Read off the page image, one by one, because the inequality directions are the entire content:
  • β = 90° → circle (Fig 10.4)
  • α < β < 90° → ellipse (Fig 10.5); strict at both ends
  • β = α → parabola (Fig 10.6)
  • 0 ≤ β < α → hyperbola (Fig 10.7); inclusive at zero, strict at α
• The parenthetical. (p. 177): for the first three cases the chapter records that the plane crosses one nappe completely. For the fourth it records that the plane goes through both. That is the sentence that explains two branches.
• Verified by inspection of the printed conditions: the four cases are mutually exclusive and, taken together, cover every β from 0 to 90°. Two of them are single values (β = 90°, β = α) and two are intervals. So a β chosen at random gives an ellipse or a hyperbola; circle and parabola sit on knife edges you have to aim for. The chapter does not say this; it falls straight out of its own list.
• Verified: β = 0 is included in the hyperbola case, so a plane parallel to the axis — but not containing it — still cuts a hyperbola. Students routinely read the range as excluding its own endpoint.

Figures to have open

• The double cone with vertex, axis, generator and both nappes named — the chapter's Fig 10.2 (p. 177). Redraw as a schematic; every later idea in the module refers back to these four words.
• One cone carrying both α and β measured against the same axis — the chapter's Fig 10.3 (p. 177). This single figure is the argument; without it the four conditions are arbitrary.
• Four cutting-plane pictures with the resulting section drawn on the cone — the chapter's Figs 10.4 to 10.7, all four of them on p. 178. Standard schematics, but they must share one cone and one viewing angle so that only the plane moves.
• A continuous β sweep. Not in the book, and the thing the book most needs: a movement is the only medium in which "one parameter, four outcomes" is visible rather than asserted.
• No photograph is needed. The portrait of Apollonius on p. 176 is decoration.

Where this sits in the book

• NCERT Mathematics, Textbook for Class XI, Chapter 10 "Conic Sections", §10.1 Introduction (p. 176), §10.2 Sections of a Cone (pp. 176–177), §10.2.1 (p. 177, with Figs 10.4–10.7 carried over to p. 178)
• Figures: Fig 10.1 (p. 176); Figs 10.2, 10.3 (p. 177); Figs 10.4–10.7 (p. 178)
• Forward pointer, deliberate: the vertex-cutting cases are §10.2.2, p. 178, covered by Cuts through the vertex, where the curve degenerates
• §10.1 (p. 176) is where the two names are credited to Apollonius, and the portrait caption on that same page carries his dates. The Historical Note (pp. 206–207) is a separate and narrower source: it records that he wrote a book on these curves, and dates that book, but says nothing about who named them