The right triangle hidden in the figure, and the single number that sets the shape

What to assume they know

• Two fixed points and a fixed total distance — Definition 4, the Note's condition, and what a, b and c stand for: half the major axis, half the minor axis, and the centre-to-focus offset
• Pythagoras' theorem, used here on a triangle whose legs are b and c
• That an ellipse is symmetric about both its axes, which the figure displays and §10.5.3 later states as an observation
• Manipulating a square root equation by squaring, and dividing one length by another to form a ratio

What they should be able to do

• Compute the sum of focal distances at a vertex of the ellipse and show it equals the major axis length
• Explain why that computation is what identifies the definition's constant as 2a
• Compute the sum of focal distances at an end of the minor axis, naming the symmetry step and the Pythagoras step separately
• Derive a² = b² + c² by equating the two computations
• Deduce that a exceeds both b and c, and hence that the major axis really is the longer of the two
• State Definition 5 and express e as a ratio of two named distances
• Express the centre-to-focus length as ae, and say why that form is useful
• Distinguish what e controls from what a controls
• Compute a, b, c and e from a given standard equation and from given foci and vertices

Where it usually goes wrong

• "a² = b² + c² is just Pythagoras on the ellipse." The right triangle in Fig 10.23 has legs b and c and hypotenuse √(b² + c²). Nothing about that triangle says the hypotenuse is a. It equals a only because the defining sum computed at Q must match the defining sum computed at P. Take the definition away and the relation vanishes.
• "c² = a² + b²." This is the hyperbola's relation, and it arrives two sections later. Confusing the two is the error to expect most often in this section. The memory hook worth teaching is structural, not verbal: in the ellipse a is the biggest of the three, so it sits alone on the left.
• "e has units, or e is a percentage." It is one length divided by another, so it is a pure number. Example 12's e = 1/2 says nothing about how large that ellipse is.
• "A bigger e means a bigger ellipse." e says nothing about size. The two ellipses in the demonstration above differ by a factor of two in every length and share an eccentricity. a sets the size; e sets the shape.
• "e can be anything positive." For an ellipse it lies strictly between 0 and 1 — but note that §10.5.2 as printed does not tell you so, so this has to be supplied. It follows immediately from 0 < c < a.
• "b is determined by nothing in particular." After §10.5.1, b is not free at all: given a and c, b is fixed. The ellipse has two independent numbers, not three, and that is what makes e a complete description of the shape.
• "The relation was assumed when 2a and 2b were named." It was not. §10.5 introduced three symbols with no relation between them, and §10.5.1 is where the relation is earned. Presenting it as a definition throws away the only proof in the section.

Questions to check understanding

• Given a and b, find c and e
• Given a and c, or given the foci and vertices, find b and e
• Given the equation in standard form, extract a, b, c and e
• Given the major axis length and the foci, find the equation's denominators
• Prove that a² = b² + c² for an ellipse, stating which points the argument uses
• Short-answer: explain why a must be larger than b

Examples worth working on the board

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

• Fig 10.23 (p. 188) — the whole topic in one drawing, and every label sits inside the artwork. Read off the page image: R is the left end of the major axis and P the right end; Q is the top end of the minor axis; F₁ and F₂ are the foci; c is marked from the centre O out to each focus; b is marked from O up to Q; a is marked from O out to P; a − c is marked from F₂ across to P; and the two slant segments from Q to each focus are both labelled √(b² + c²). That last labelling is the Pythagoras step done for the reader inside the figure.
• The vertex computation (§10.5.1, p. 188). Take P at the right-hand end of the major axis. Then F₁P splits as F₁O + OP and F₂P is the remainder. Verified by an added reading of the figure: F₁O = c, OP = a, and F₂P = a − c, so the sum is (c + a) + (a − c) = 2a. The c cancels, which is why a vertex is the right point to start at.
• What that computation actually settles. Definition 4 called the sum "a constant" and gave it no value. §10.5 then wrote 2a for the major axis, on its own authority. This computation is the bridge: it shows the definition's constant is the major axis length. Verified as load-bearing: without it, 2a in the standard derivation of §10.5.3 would be an unjustified substitution.
• The minor-axis computation (§10.5.1, p. 188). Take Q at the top of the minor axis. Triangle OQF₂ is right-angled at O with legs OQ = b and OF₂ = c, so QF₂ = √(b² + c²), and by the mirror symmetry across the minor axis QF₁ is the same length. The sum is therefore 2√(b² + c²).
• Equating them (p. 188). Both sums equal the same defining constant, so 2√(b² + c²) = 2a. Verified: dividing by 2 and squaring gives a² = b² + c², and rearranging gives c = √(a² − b²). The chapter prints both forms.
• The three consequences — derived here, not printed in §10.5.1. Verified: since c > 0 for a genuine ellipse, a² > b², so a > b. Since b > 0, a² > c², so a > c. Therefore the axis through the foci is the longer axis, which retrospectively justifies the name "major" that §10.5 had already used.
• Definition 5 (§10.5.2, p. 188). The eccentricity e divides how far the centre lies from a focus by how far it lies from a vertex, so e = c/a. The chapter then records that a focus stands ae away from the centre — which is c = ae rearranged, and is the form §10.5.4 will need.
• The range of e — supplied here. Verified: 0 < c < a gives 0 < e < 1 for every ellipse with distinct foci. §10.5.2 does not print this range — checked on the p. 188 page image, where the subsection consists of Definition 5 and the ae remark and nothing else. A student who has read only §10.5.2 has not been told what values e can take.
• Example 9 (pp. 192–193). The ellipse x²/25 + y²/9 = 1. Verified: a = 5, b = 3, c = √(25 − 9) = 4, so e = 4/5, foci (±4, 0), vertices (±5, 0), major axis 10, minor axis 6.
• Example 10 (p. 193). The ellipse 9x² + 4y² = 36. Verified: dividing by 36 gives x²/4 + y²/9 = 1, so here a = 3 and b = 2 with the major axis vertical; c = √(9 − 4) = √5 and e = √5/3, foci (0, ±√5), vertices (0, ±3).
• Example 11 (pp. 193–194). Vertices (±13, 0), foci (±5, 0). Verified: a = 13 and c = 5, so b² = 169 − 25 = 144 and b = 12; e = 5/13.
• Example 12 (p. 194). Major axis 20, foci (0, ±5). Verified: a = 10 and c = 5, so b² = 100 − 25 = 75; e = 1/2. A good case for showing that e is a pure number while a and b carry the size.
• A scale-free demonstration worth building. Verified: the ellipses x²/25 + y²/9 = 1 and x²/100 + y²/36 = 1 have a = 5, b = 3 and a = 10, b = 6, so c = 4 and c = 8 and both have e = 4/5. Doubling every length leaves e alone. The chapter never sets two such ellipses side by side, and it is the cheapest way to show what e measures.

Figures to have open

• Fig 10.23 (p. 188) redrawn with every label carried over — the two vertices, the minor-axis end, both foci, the centre, and the lengths a, b, c, a − c and the two slant segments. This is the chapter's single densest figure and the argument cannot be made without it.
• The right triangle O Q F₂ pulled out of Fig 10.23 and shown on its own with legs b and c. Not a printed figure, and the step most likely to be skated over.
• A two-panel comparison of ellipses with the same e and different a. Not printed, and the only way to make "e is scale-free" visible rather than asserted.
• A slider figure sweeping e from near 0 to near 1 at fixed a, showing the foci migrating outwards and the curve flattening. Not printed; it is what makes Definition 5 mean something.

Where this sits in the book

• NCERT Mathematics, Textbook for Class XI, Chapter 10 "Conic Sections", §10.5.1, whose heading names the relationship it establishes between the two semi-axes and the focal offset (p. 188); §10.5.2 Eccentricity, including Definition 5 (p. 188)
• Figure: Fig 10.23 (p. 188)
• Examples 9 and 10 (pp. 192–193); Examples 11 and 12 (pp. 193–194)
• Deliberate backward reference inside the chapter: the definition and the naming of a, b, c are §10.5, p. 187 — Two fixed points and a fixed total distance
• Deliberate forward reference inside the chapter: the relation derived here is the substitution that tidies the standard derivation at §10.5.3, p. 190, and the form c = ae is what §10.5.4 needs on p. 192
• The chapter's Summary (p. 205) restates Definition 5 but prints neither a² = b² + c² nor any range for e