Conic Sections

2.1 Core concepts

• A conic section is a curve obtained by intersecting a right circular cone (double-napped) with a plane; the type depends on the angle β the plane makes with the vertical axis and the semi-vertical angle α of the cone (NCERT §10.2, p. 176–177). The Greek mathematician Apollonius (~200 BCE) first studied these systematically.
• Sections (non-degenerate, plane not through vertex): β = 90° → circle; α < β < 90° → ellipse; β = α → parabola; 0 ≤ β < α → hyperbola (plane cuts both nappes) (NCERT §10.2.1, p. 177).
• Degenerated conics (plane through vertex): α < β ≤ 90° → a point; β = α → a straight line (degenerate parabola); 0 ≤ β < α → a pair of intersecting straight lines (degenerate hyperbola) (NCERT §10.2.2, p. 178). These correspond to "limit" cases when the plane just touches the vertex.
• Circle: locus of points in a plane equidistant from a fixed point (centre); equation with centre (h, k) and radius r is (x – h)² + (y – k)² = r²; with centre at origin it reduces to x² + y² = r² (NCERT §10.3, p. 179–180). Eccentricity of circle is 0 (a limiting case of ellipse).
• The general form x² + y² + 8x + 10y – 8 = 0 is reduced by completing the square to (x + 4)² + (y + 5)² = 49, giving centre (–4, –5) and radius 7 — the standard NCERT technique for any general circle equation (NCERT Example 3, p. 180). Any equation Ax² + Ay² + Bx + Cy + D = 0 (A ≠ 0) represents a circle.
• Parabola: set of all points in a plane equidistant from a fixed line (directrix) and a fixed point (focus) not on the line; the line through the focus perpendicular to the directrix is the axis; intersection of parabola with the axis is the vertex (NCERT §10.4, p. 182). Eccentricity of parabola is 1.
• Four standard parabolas (vertex at origin, axis on a coordinate axis): y² = 4ax (opens right), y² = –4ax (opens left), x² = 4ay (opens upward), x² = –4ay (opens downward); a y² term ⇒ axis along x-axis, an x² term ⇒ axis along y-axis (NCERT §10.4.1, p. 183–184).
• Latus rectum of a parabola is the chord through the focus perpendicular to the axis with endpoints on the curve; its length for y² = 4ax is 4a (NCERT §10.4.2, p. 185).
• Ellipse: set of points the sum of whose distances from two fixed points (foci) is a constant 2a, with 2a greater than the distance 2c between foci (NCERT §10.5, p. 187).
• Ellipse geometry: length of major axis 2a, minor axis 2b, distance between foci 2c, and the relation c² = a² – b² (equivalently a² = b² + c²) (NCERT §10.5.1, p. 188). The square-root relation arises from the right triangle B–C–F where B is on the minor axis.
• Eccentricity of ellipse: e = c/a; since c < a, we have e < 1; focus lies at distance ae from the centre (NCERT §10.5.2, p. 188). The closer e is to 0, the more circular; the closer to 1, the more elongated.
• Standard equations of ellipse: with centre at origin, x²/a² + y²/b² = 1 (foci on x-axis) and x²/b² + y²/a² = 1 (foci on y-axis); the foci always lie on the major axis — i.e., on the axis whose square has the larger denominator (NCERT §10.5.3, p. 189–191).
• Latus rectum of ellipse: the chord through a focus perpendicular to the major axis with endpoints on the ellipse; its length is 2b²/a (NCERT §10.5.4, p. 192).
• Hyperbola: set of points the difference of whose distances from two fixed foci is a constant 2a; mid-point of foci = centre; line through foci = transverse axis; perpendicular through centre = conjugate axis; intersections with transverse axis = vertices (NCERT §10.6, p. 195–196).
• Hyperbola geometry: distance between foci 2c, length of transverse axis 2a, length of conjugate axis 2b, with b = √(c² – a²) so that c² = a² + b² (NCERT §10.6, p. 196). Note the sign difference from the ellipse relation.
• Eccentricity of hyperbola: e = c/a; since c ≥ a we always have e ≥ 1; foci are at distance ae from centre (NCERT §10.6.1, p. 197).
• Standard equations of hyperbola (centre at origin): x²/a² – y²/b² = 1 (transverse axis along x-axis) and y²/a² – x²/b² = 1 (transverse axis along y-axis); the positive term's denominator gives the transverse axis (NCERT §10.6.2, p. 197–199).
• An equilateral hyperbola is one in which a = b (NCERT §10.6.2 Note, p. 199). Its eccentricity is √2.
• Latus rectum of hyperbola: chord through a focus perpendicular to the transverse axis; its length is 2b²/a — identical formula to the ellipse (NCERT §10.6.3, p. 200).
• The four conics interrelate: circle is a special ellipse with e = 0 (a = b); parabola is the limit of an ellipse (or hyperbola) as e → 1; hyperbola has e > 1. The unified focus–directrix definition gives all four as "loci of constant ratio of distances", with the ratio e determining the type.
• Asymptotes of the standard hyperbola x²/a² − y²/b² = 1 are the lines y = ±(b/a) x; these are the diagonals of the central rectangle of sides 2a × 2b and serve as guidelines for sketching the hyperbola accurately.
• Practical curve-sketching: for the ellipse, mark vertices (±a, 0), co-vertices (0, ±b), foci (±c, 0); for the hyperbola, mark vertices (±a, 0), foci (±c, 0), and asymptotes; for the parabola, mark vertex (0, 0), focus (a, 0), and directrix x = −a.
• Every conic can be written in polar coordinates centred at a focus as r = ed/(1 + e cos θ), where d is the distance from focus to directrix; this unified treatment is studied in advanced courses.
• Conic sections appear extensively in physics (Kepler orbits — ellipses, parabolic trajectories, hyperbolic comet paths) and engineering (parabolic mirrors, elliptical gears).
• Quick-recognition tip for CUET MCQs: see y² = ___ x ⇒ parabola opening along x-axis; x² + y² = ___ ⇒ circle; positive coefficients on both x² and y² with unequal denominators ⇒ ellipse; one positive, one negative ⇒ hyperbola.
• Important historical note: Pappus of Alexandria proved (around 320 CE) the unified focus–directrix theorem; Kepler used the ellipse in his first law of planetary motion (1609); Newton's gravitational theory rigorously derives Keplerian orbits as conics.

2.2 Definitions to memorise

2.3 Diagrams / processes to remember

• Fig 10.1–10.3 (p. 176–177): line m rotated about axis l at fixed angle α generates the double-napped cone; β is the angle the cutting plane makes with the vertical axis.
• Fig 10.4–10.7 (p. 178): the four non-degenerate cases — circle (β = 90°), ellipse (α < β < 90°), parabola (β = α), hyperbola (0 ≤ β < α, cuts both nappes).
• Fig 10.8–10.10 (p. 179): degenerated conics — point, single line, pair of intersecting straight lines (plane through vertex).
• Fig 10.11–10.12 (p. 180): circle centred at origin and at (h, k).
• Fig 10.15 (a)–(d) (p. 183): the four standard parabolas y² = 4ax, y² = –4ax, x² = 4ay, x² = –4ay.
• Fig 10.17–10.18 (p. 185): construction showing the latus rectum length 4a for y² = 4ax.
• Fig 10.21–10.23 (p. 187–188): ellipse — vertices, major/minor axis, the right-triangle picture that yields a² = b² + c².
• Fig 10.27–10.30 (p. 196–198): hyperbola — vertices on transverse axis, b = √(c² – a²), two standard orientations.
• Process — reduce general circle to standard form: group x-terms and y-terms; complete squares; rearrange to (x − h)² + (y − k)² = r² form; read off centre and radius.
• Process — identify parabola: check which variable is squared (y² ⇒ horizontal axis; x² ⇒ vertical axis); check sign of the linear coefficient (positive ⇒ opens right/up; negative ⇒ left/down); compute a = (|coefficient|)/4 and locate focus at (a, 0) or (0, a).
• Process — find focus/eccentricity of ellipse: read a², b² from denominators (a > b); compute c = √(a² − b²); foci at (±c, 0) if x²/a² + y²/b² with a > b; e = c/a; latus rectum = 2b²/a.
• Process — find focus/eccentricity of hyperbola: read a² (positive term), b² (negative term); compute c = √(a² + b²); e = c/a; latus rectum = 2b²/a.

2.4 Common confusions / NTA trap points

• c² sign flip: for an ellipse c² = a² – b² (so b < a along major axis), but for a hyperbola c² = a² + b². Mixing the two is the most common slip.
• Which axis carries the foci? For an ellipse the foci lie on the major axis — i.e., the axis whose square has the larger denominator (p. 191). For a hyperbola the foci lie on the transverse axis — i.e., the axis of the positive term in x²/a² – y²/b² = 1 (p. 200). NTA distractors swap these rules.
• Direction of opening for parabola: y² = 4ax opens right, y² = –4ax opens left, x² = 4ay opens up, x² = –4ay opens down (p. 184). Misreading the sign or coordinate puts foci on the wrong axis.
• Eccentricity bounds: e < 1 for ellipse, e = 1 conceptually corresponds to the parabola case (β = α), e > 1 for hyperbola. Confusing "e ≤ 1" with "e < 1" is a trap.
• Latus-rectum formula reuse: 2b²/a is the latus-rectum length for both ellipse and hyperbola — but the underlying b² differs because c² = a² – b² vs c² = a² + b². Plugging the wrong relation gives a wrong b² and a wrong latus rectum.
• Confusing focus and directrix: the focus is a point, the directrix is a line. The parabola is the locus equidistant from a point and a line; ellipse and hyperbola are loci involving distances to two points.
• Vertex vs centre: the parabola has a single vertex; the ellipse and hyperbola each have two vertices and one centre.
• Forgetting to compare a and b for ellipse: if a < b in the equation x²/a² + y²/b² = 1, then the major axis is along the y-axis, not the x-axis.
• Confusing 4a (parabola) with 2b²/a (ellipse/hyperbola): the latus rectum formula differs by type.
• Mis-identifying the type from a general equation: look for Ax² + Cy² + …; A = C ⇒ circle (after appropriate sign); A, C same sign different magnitudes ⇒ ellipse; A, C opposite signs ⇒ hyperbola; one of A, C zero ⇒ parabola.
• Mistreating a², b² as algebraic quantities that can be negative: they are always positive squares.
• Confusing the focal chord with latus rectum: latus rectum is the specific focal chord perpendicular to the axis.

2.5 Key formulas & theorems

2.6 Solved examples (NCERT-grounded)

Example A (NCERT Example 3, p. 180). Centre, radius of x² + y² + 8x + 10y − 8 = 0.

Step 1 — complete squares: (x + 4)² − 16 + (y + 5)² − 25 − 8 = 0. Step 2 — rearrange: (x + 4)² + (y + 5)² = 49. Step 3 — read off: centre (−4, −5), radius 7.

Example B (NCERT Example 5, p. 185). Focus and latus rectum of y² = 8x.

Step 1 — compare with y² = 4ax: 4a = 8 ⇒ a = 2. Step 2 — focus: (a, 0) = (2, 0). Step 3 — latus rectum: 4a = 8.

Example C (NCERT Example 7, p. 186). Equation of parabola with vertex (0,0), focus (0, 2).

Step 1 — focus on y-axis: axis along y-axis; form x² = 4ay with a = 2. Step 2 — substitute: x² = 4·2·y. Step 3 — final: x² = 8y.

Example D (NCERT Example 9, p. 192). Eccentricity, latus rectum of x²/25 + y²/9 = 1.

Step 1 — read a, b: a = 5, b = 3. Step 2 — c = √(a² − b²): √(25 − 9) = 4. Step 3 — e and LR: e = c/a = 4/5; LR = 2b²/a = 18/5.

Example E (NCERT Example 14(i), p. 200). Foci, e, LR of x²/9 − y²/16 = 1.

Step 1 — a, b: a = 3, b = 4. Step 2 — c: √(9 + 16) = 5. Step 3 — read off: foci (±5, 0), e = 5/3, LR = 32/3.