Balancing a point against a line, and the four equations that result

What to assume they know

• Fixed distance from a fixed point, turned into an equation — the distance formula used as a definition-translator, and the habit of putting a figure on axes before writing anything
• The distance from a point to a vertical line, read as a horizontal difference
• That √(t²) is the non-negative root, and that dropping the square root needs the inside to be non-negative — the step the chapter performs without comment
• Expanding and cancelling squared binomials
• One surface, four curves, chosen by the angle of the cut — the parabola as the knife-edge section β = α, which is why an unbounded curve is expected here

What they should be able to do

• State Definition 2 and name its two ingredients and the exclusion it carries
• Identify focus, directrix, axis and vertex on a drawing
• Explain the degenerate case that arises when the focus is allowed onto the directrix
• Set up the coordinate frame the chapter uses, and say what each choice buys
• Derive y² = 4ax from the definition, naming the point at which the square root is cleared
• Reproduce the converse argument and say why the forward argument alone is not a proof of equality
• Explain, from a > 0, why the curve occupies only two quadrants
• Write down the other three standard equations and say which orientation each describes
• Read the axis of symmetry and the opening direction off a given equation
• Fit a standard parabola to a stated condition — a focus, a directrix, a vertex, or a point it passes through

Where it usually goes wrong

• "a is the coefficient." a is a length — the distance from the vertex to the focus, and equally from the vertex to the directrix. The distance from directrix to focus is 2a and the coefficient in the equation is 4a. Three different numbers, one letter, and students conflate them constantly.
• "The four standard equations are four different curves." One curve, four placements of the axes. The chapter derives one and asserts three, which is only honest because the other three are the same argument with the letters permuted and signs flipped.
• "y = x² is the standard parabola." Not in this chapter's sense. Every standard form here has the vertex at the origin and the focus on an axis, and is written with the squared variable alone on the left and 4a as the coefficient. y = x² is x² = 4ay with 4a = 1, and the chapter's parameter is a, not the coefficient.
• "The equation is the parabola, obviously." That is exactly what the converse paragraph on p. 184 exists to establish, and it is not obvious: the forward argument only shows the curve sits inside the solution set. Skipping the converse is skipping the proof.
• "A y² term puts the mirror line along the y-axis." The opposite. y appearing squared means +y and −y give the same x, so the mirror is the x-axis. The printed observations say this; students routinely read the letter instead of the argument.
• "The directrix touches the curve." It never meets it. Nor does the focus lie on the curve; Definition 2 explicitly excludes the focus from the directrix, and the vertex sits halfway between them.
• "Deflection measured downwards can be substituted as-is." Miscellaneous Example 18 is built to punish this. Read Fig 10.32's marked distances before writing any coordinate.
• "Any focus and any directrix can be handled by these four forms." The Note on p. 184 says otherwise and declares the general case out of scope. A brief that implies the four forms are complete is teaching a false closure.

Questions to check understanding

• Given the equation, state focus, axis, directrix and opening direction
• Given a focus and a directrix, write the standard equation
• Given a vertex and a focus, write the standard equation
• Given a vertex at the origin, an axis, and a point on the curve, find the equation — including cases where a is a fraction
• Explain why the curve lies in only two quadrants when a > 0
• Physical-setting problems: reflector depth and diameter, arch height and width, suspension cable, beam deflection
• Short-answer: why does the chapter prove the converse, and what would be missing without it

Examples worth working on the board

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

• Fig 10.13 (p. 182). The parabola with its directrix drawn as the line l on the left and the focus F on the right. Three points are marked on the curve and three on the directrix, and the three equalities of focal distance against directrix distance are printed beneath the figure. Read off the page image: the labelling is inside the artwork.
• Fig 10.14 (p. 182). The same curve with directrix, focus, axis and vertex each named. The axis is drawn horizontal and the vertex sits between the directrix and the focus — which is Fig 10.16's construction stated in advance.
• The Note on the degenerate case (§10.4, p. 182). Put the fixed point on the fixed line and the set of equidistant points is the line through that point perpendicular to the fixed line. This is a second and quite separate route to a degenerate parabola; the cone route is §10.2.2 case (b), covered in Cuts through the vertex, where the curve degenerates.
• The frame (§10.4.1, p. 183, Fig 10.16). F is the focus, l the directrix, FM the perpendicular from F to l, and O the midpoint of FM. O becomes the origin, the line MO extended becomes the x-axis, the perpendicular to it at O becomes the y-axis. The focus and the directrix are set 2a apart. So the focus is (a, 0), the directrix is x + a = 0, and O is on the curve by the definition — which is what makes O the vertex rather than an arbitrary point.
• The forward derivation (p. 183). For P(x, y) on the curve, the foot of the perpendicular to the directrix is B(−a, y). Setting PF equal to PB and squaring gives (x − a)² + y² = (x + a)², and the x² and a² terms cancel, leaving y² = 4ax with a > 0. Verified: the cancellation removes everything except ∓2ax, so the whole derivation is one subtraction once the squares are expanded.
• The converse (p. 184). Substituting y² = 4ax back into PF gives (x − a)² + 4ax, and verified by an added expansion, x² − 2ax + a² + 4ax is x² + 2ax + a², which is (x + a)². So PF equals PB and the point is on the parabola. This is the step that turns "the curve satisfies the equation" into "the equation is the curve".
• Discussion (p. 184). Since a > 0 and y² = 4ax, x cannot be negative; it may be zero or positive. Verified: the curve therefore lives in the first and fourth quadrants, extends without bound, and its axis is the positive x-axis.
• The other three orientations (Fig 10.15 (b), (c), (d), pp. 182–183). Read off the figure captions and the drawings themselves: (b) has focus (−a, 0) and directrix x = a, equation y² = −4ax; (c) has focus (0, a) and directrix y = −a, equation x² = 4ay; (d) has focus (0, −a) and directrix y = a, equation x² = −4ay. Cite these as Fig 10.15; see Notes.
• The three printed observations (p. 184). A y² term puts the axis of symmetry along the x-axis; an x² term puts it along the y-axis. With the axis along the x-axis the curve opens right when x carries a positive coefficient and left when that coefficient is negative; with the axis along the y-axis it opens upwards or downwards by the sign of the coefficient of y.
• The Note bounding the section (p. 184). The standard equations all require the focus on a coordinate axis and the vertex at the origin, so the directrix ends up parallel to the other axis. A parabola with an arbitrary focus and an arbitrary directrix is declared out of scope.
• Example 6 (p. 186). Focus (2, 0), directrix x = −2. Verified: the focus is on the x-axis and to the right of the directrix, so the form is y² = 4ax with a = 2, giving y² = 8x.
• Example 7 (p. 186). Vertex (0, 0), focus (0, 2). Verified: the form is x² = 4ay with a = 2, giving x² = 8y.
• Example 8 (p. 186). A parabola whose mirror line is the y-axis, passing through (2, −3). Verified: the point is in the fourth quadrant so the curve must open downwards, giving x² = −4ay; substituting, 4 = 12a, so a = 1/3 and the equation is 3x² = −4y. The reasoning that fixes the sign before any arithmetic is the content of this example.
• Exercise 10.2 items 7–12 (pp. 186–187): focus (6, 0) with directrix x = −6; focus (0, −3) with directrix y = 3; vertex (0, 0) with focus (3, 0); vertex (0, 0) with focus (−2, 0); vertex (0, 0) through (2, 3) with axis along the x-axis; vertex (0, 0) through (5, 2) and symmetric about the y-axis. Verified, items 11 and 12: item 11 gives 9 = 8a so a = 9/8 and the equation is 2y² = 9x; item 12 gives 25 = 8a so a = 25/8 and the equation is 2x² = 25y. These two are the only items in the set where a is not a whole number, and they are the ones that show a is a length, not a tidy coefficient.
• Miscellaneous Example 18 (pp. 202–203, Fig 10.32). A beam 12 m between supports, sagging 3 cm at the centre, the sag shaped as a parabola with vertex at the lowest point and axis vertical. Asked: at what horizontal distance from the midpoint has the beam sagged only 1 cm below its supports. Verified, and this is the trap: the figure marks 3 cm from the support line down to the vertex, 1 cm down from the support line to the point B, and 2 cm from B down to the vertex. So a deflection of 1 cm means the height above the vertex is 2 cm, not 1 cm. Working in metres, the curve passes through (6, 3/100) so 36 = 4a(3/100) and 4a = 1200; then x² = 1200 × 2/100 = 24 and x = 2√6 m. An explanation that reads "1 cm" straight into the equation gets the wrong answer, and the only thing that prevents it is Fig 10.32's three marked distances.
• Miscellaneous Exercise item 2 (p. 204). A parabolic arch, axis vertical, standing 10 m tall and spanning 5 m across its base; asked for its width at a height 2 m down from the vertex. Verified: with the vertex at the top and the base 10 m below, the curve passes through (2.5, −10), so 6.25 = 40a and 4a = 0.625; at 2 m below the vertex, x² = 0.625 × 2 = 1.25, so the half-width is √1.25 and the full width is √5, about 2.24 m.
• Miscellaneous Exercise item 3 (p. 204). A suspension bridge cable hanging as a parabola; roadway horizontal and 100 m long; longest wire 30 m, shortest 6 m; find the wire 18 m from the middle. Verified: the vertex sits 6 m above the roadway, so at x = 50 the cable is 24 m above the vertex, giving 2500 = 4a(24) and 4a = 625/6. At x = 18 the height above the vertex is 324 × 6/625 = 3.1104, so the wire is about 9.11 m.
• Miscellaneous Exercise item 8 (p. 204). An equilateral triangle inscribed in y² = 4ax with one vertex at the parabola's vertex. Verified: by symmetry the other two are (x, ±y) with y = x tan 30°, so x²/3 = 4ax gives x = 12a and y = 4√3 a, and the side is 8√3 a.

Figures to have open

• Fig 10.13 (p. 182) redrawn with all three equal-distance pairs drawn as marked segments. The printed figure states the equalities in text beside the drawing; a movement should show one pair moving along the curve while both lengths stay equal, which is the definition in motion.
• Fig 10.14 (p. 182) redrawn with the four parts named. Standard schematic.
• Fig 10.16 (p. 183) redrawn as the construction it is: focus, directrix, the perpendicular FM, the midpoint O, then the axes appearing. The printed figure shows the finished frame; the explanation needs the order.
• Fig 10.15 (a) to (d) (pp. 182–183) as a 2x2 grid on one shared origin, each panel carrying its focus, its directrix and its equation. This is the topic's single most useful still.
• Fig 10.32 (p. 203) redrawn with all three distances marked — 3 cm, 1 cm and 2 cm. Without the 2 cm the worked example is unintelligible.
• A shaded half-plane for section 8. Standard schematic.

Where this sits in the book

• NCERT Mathematics, Textbook for Class XI, Chapter 10 "Conic Sections", §10.4 Parabola (p. 182) including Definition 2 and the Note; §10.4.1 Standard equations of parabola (pp. 182–184)
• Examples 6, 7 and 8 (p. 186); Exercise 10.2 items 7–12 (pp. 186–187)
• Miscellaneous Example 18 (pp. 202–203) with Fig 10.32; Miscellaneous Exercise on Chapter 10 items 2, 3 and 8 (p. 204)
• Figures: Fig 10.13 and Fig 10.14 (p. 182); Fig 10.15 (a) and (b) (p. 182), Fig 10.15 (c) and (d) and Fig 10.16 (p. 183); Fig 10.32 (p. 203)
• Deliberate reference inside the chapter: the parabola's place among the sections of a cone is §10.2.1, p. 177, and its degenerate cone section is §10.2.2 case (b), p. 178
• The chapter's Summary (p. 205) restates Definition 2 and y² = 4ax only, and prints none of the other three standard forms