Two fixed points and a fixed total distance

What to assume they know

• Fixed distance from a fixed point, turned into an equation — the circle as a locus defined by a distance condition, and the habit of reading a definition as a membership test
• The triangle inequality, in the form that any two sides of a triangle together exceed the third, and that equality means the triangle has flattened
• One surface, four curves, chosen by the angle of the cut — the ellipse as the section for α < β < 90°, which is why a bounded curve is expected here
• Reading a strict inequality and knowing what its boundary case looks like

What they should be able to do

• State Definition 4 and name its two ingredients and the quantity held constant
• State the Note's condition relating the constant to the distance between the foci
• Justify that condition from the triangle inequality rather than accepting it
• Describe what the locus becomes when the constant equals the focal separation, and when it is smaller
• Identify centre, major axis, minor axis, foci and vertices on a drawing
• Explain why the foci lie on the major axis by definition rather than by proof
• Use the symbols 2a, 2b and 2c correctly, and say which quantity each names
• Apply the definition directly to a stated physical situation and check the condition before computing anything
• Say what §10.5 has established and what it has deliberately not yet established

Where it usually goes wrong

• "Any two points and any positive constant give an ellipse." They do not. The constant must exceed the distance between the points, and the Note on p. 187 says so. This is the single most-skipped sentence in the section.
• "The Note is a fact about ellipses you have to remember." It is a consequence of the triangle inequality, which students already know. Deriving it takes fifteen seconds and turns a memorised clause into an obvious one.
• "An ellipse is a squashed circle." It looks like one, and the two definitions are genuinely different. The circle's definition names one point; the ellipse's names two and a sum. The circle appears in this family when the two foci come together, and §10.5 does not discuss that case at all.
• "The constant is a." The constant is 2a — the whole major axis, not half of it. Every subsequent derivation writes it as 2a and students who stored a get every sign wrong downstream.
• "b measures centre to focus." That is c. b runs from the centre out to an end of the minor axis. Fig 10.22 marks both, which is why it must be redrawn rather than described.
• "The foci are on the ellipse." They are inside it. Nothing in the definition puts them on the curve, and Fig 10.20 draws them in the interior.
• "The major axis is the longer one, by definition." Not in this chapter. §10.5 defines the major axis as the one through the foci. That it is also the longer one is a consequence of the relation proved in §10.5.1, and it has not been established at this point in the book. Naming it "major" before proving it is longer is the chapter getting slightly ahead of itself.

Questions to check understanding

• State the definition of an ellipse and the condition on its constant
• Given two foci and a constant, decide whether an ellipse exists
• Identify centre, foci, vertices, major and minor axes on a diagram
• Given the major axis length and the focal separation, write down a and c
• Recognise a described physical situation as an ellipse and check admissibility
• Short-answer: what is the locus when the constant equals the distance between the foci

Examples worth working on the board

Values marked verified are worked out here from the chapter's printed data.

• Fig 10.20 (p. 187). An ellipse with the two foci marked F₁ and F₂ on the interior, three points P₁, P₂ and P₃ marked on the curve, and all six connecting segments drawn. Beneath the figure the chapter prints the chain of equalities saying that all three sums are the same. Read off the page image; the labelling is inside the artwork. This figure asserts the constancy; it does not derive it, because constancy is the definition.
• The Note (§10.5, p. 187, in a tinted box). Read off the page image, one sign at a time: the fixed total must always exceed how far apart the two foci are. The comparison is strict, not inclusive.
• Why the Note holds — supplied by this brief, not by the chapter. For a point P off the line through the foci, P, F₁ and F₂ form a genuine triangle, so PF₁ + PF₂ exceeds F₁F₂. Writing the sum as 2a and the separation as 2c, that is 2a > 2c, so a > c. Verified as sharp: equality happens exactly when P lies on the segment F₁F₂, at which point the triangle has flattened to a segment.
• The two boundary cases — also supplied here. If the constant is set equal to the separation, every point of the segment F₁F₂ satisfies the condition and no point off it does, so the locus is that segment. If the constant is set below the separation, no point in the plane satisfies it and the locus is empty. Neither case is printed in §10.5; both are worth ten seconds, because they are what the Note is protecting against.
• Fig 10.21 (p. 187). The named ellipse: A and B at the ends of the major axis and labelled as vertices, C and D at the ends of the minor axis, O marked Centre, F₁ and F₂ marked on the major axis, and the major and minor axes both named with arrows. All labels are inside the artwork.
• Fig 10.22 (p. 187), measured on the printed page. The same ellipse carrying lengths instead of names: c marked twice, a marked twice, b marked twice, each pair reading out from the centre. This is where 2a, 2b and 2c are pinned to the picture.

Where the six marks actually sit matters for the redraw. Not one of them is drawn on the axis it measures. The pair of c marks runs as a dashed span above the curve, the pair of a marks as a dashed span below it, and the pair of b marks as a dashed span off to the right, each tied back to the feature it belongs to by dashed leader lines. Moving them onto the axes would tidy the figure into something the page does not show, and a student holding the book would not recognise it.

• The three lengths (§10.5, p. 187). Major axis 2a, minor axis 2b, distance between the foci 2c; semi-major a, semi-minor b. Verified as the honest state of play at the end of §10.5: three symbols have been introduced and no relation among them has been stated. The relation is §10.5.1's business, and an explanation that reaches for a² = b² + c² here is borrowing from the next topic.
• Miscellaneous Exercise item 7 (p. 204). A runner on a racecourse for whom the sum of the distances to two flag posts is always 10 m, the posts being 8 m apart; find the equation of the path. Verified, and the order of work is the lesson: first check 10 > 8, so the Note's condition holds and a path exists; then 2a = 10 gives a = 5 and 2c = 8 gives c = 4. Turning that into an equation needs the relation from The right triangle hidden in the figure, and the single number that sets the shape and the standard form from Putting the centre at the origin, and reading the axes off the equation; the part that belongs to this topic is recognising the situation as Definition 4 and checking it is admissible.
• A counterexample worth inventing. Same two posts 8 m apart, but a runner whose distances always total 6 m. Verified: no such runner exists, because 6 < 8. Posing it beside item 7 is the cheapest way to show the Note doing work.

Figures to have open

• Fig 10.20 (p. 187) redrawn, and preferably shown step by step: a point travelling round the ellipse with both focal segments drawn and a live readout of their sum holding steady. This is the definition made visible, and a still cannot do it.
• Fig 10.21 (p. 187) redrawn with centre, both axes, both vertices and both foci named. Standard schematic.
• Fig 10.22 (p. 187) redrawn with a, b and c each marked twice, out from the centre, and each pair kept off the axis it measures on its own dashed span with leader lines, as printed — c above the curve, a below it, b to the right. Keeping the "twice" is what stops students storing a when they mean 2a.
• A triangle P F₁ F₂ with the two focal segments and the base drawn, for section 4. Not a printed figure.
• A three-panel sequence — constant above, equal to, and below the focal separation — showing an ellipse, a segment and an empty frame. Not printed, and the visual payoff of the whole topic.

Where this sits in the book

• NCERT Mathematics, Textbook for Class XI, Chapter 10 "Conic Sections", §10.5 Ellipse (p. 187): Definition 4, the Note, and the naming of centre, major axis, minor axis, vertices and the lengths 2a, 2b, 2c
• Figures: Fig 10.20, Fig 10.21 and Fig 10.22, all on p. 187
• Miscellaneous Exercise on Chapter 10 item 7 (p. 204)
• Deliberate reference inside the chapter: the ellipse's place among the sections of a cone is §10.2.1, p. 177
• Deliberate forward reference inside the chapter: the relation among a, b and c is §10.5.1, p. 188 — The right triangle hidden in the figure, and the single number that sets the shape
• The chapter's Summary (p. 205) restates Definition 4 and nothing else from this section; the Note's condition does not appear there