Lesson 12 of 12 · 15 min
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Circle, parabola, ellipse and hyperbola with NCERT
Must-know facts
18 facts
- 01β = 90°: circle; α < β < 90°: ellipse; β = α: parabola; 0 ≤ β < α: hyperbola.
- 02Plane through the vertex: a point, a line (degenerate parabola) or a pair of lines (degenerate hyperbola).
- 03Circle with centre (h, k), radius r: (x − h)² + (y − k)² = r².
- 04x² + y² + 8x + 10y − 8 = 0 has centre (−4, −5) and radius 7.
- 05Parabola: distance from focus = distance from directrix.
- 06y² = 4ax: focus (a, 0), directrix x = −a, axis the x-axis, latus rectum 4a.
- 07y² = 8x: focus (2, 0), directrix x = −2, latus rectum 8.
- 08Ellipse: PF₁ + PF₂ = 2a, with a² = b² + c² and e = c/a < 1.
- 09x²/a² + y²/b² = 1 (a > b): foci (±c, 0), vertices (±a, 0).
- 10The larger denominator decides the major axis of an ellipse.
- 11x²/25 + y²/9 = 1: foci (±4, 0), e = 4/5, latus rectum 18/5.
- 12Latus rectum of an ellipse or a hyperbola: 2b²/a.
- 13Hyperbola: |PF₁ − PF₂| = 2a, with c² = a² + b² and e = c/a > 1.
- 14The positive term decides the transverse axis of a hyperbola.
- 15x²/9 − y²/16 = 1: foci (±5, 0), vertices (±3, 0), e = 5/3, latus rectum 32/3.
- 16a = b gives an equilateral hyperbola.
- 17Parabolic mirror, focus 5 cm, depth 45 cm: rim 60 cm across.
- 18Rod of 15 cm with AP = 6 cm traces x²/81 + y²/36 = 1.
Common traps
Where marks are lost
Formulas
10 to know
Circle
(x − h)² + (y − k)² = r²
Centre (h, k), radius r; x² + y² = r² when the centre is the origin.
Parabola (opens right)
y² = 4ax
Focus (a, 0), directrix x = −a, latus rectum 4a.
Other parabolas
y² = −4ax, x² = 4ay, x² = −4ay
Open left, up, down; a > 0.
Ellipse (major axis on x)
x²/a² + y²/b² = 1, a > b
Foci (±c, 0), vertices (±a, 0).
Ellipse (major axis on y)
x²/b² + y²/a² = 1, a > b
Foci (0, ±c), vertices (0, ±a).
Ellipse relation
c² = a² − b², e = c/a < 1
Foci are ae from the centre.
Hyperbola (transverse on x)
x²/a² − y²/b² = 1
Foci (±c, 0), vertices (±a, 0).
Hyperbola (transverse on y)
y²/a² − x²/b² = 1
Foci (0, ±c), vertices (0, ±a).
Hyperbola relation
c² = a² + b², e = c/a > 1
a = b gives an equilateral hyperbola.
Latus rectum
Parabola 4a; ellipse and hyperbola 2b²/a
Key terms
14 terms
- Double napped cone
- The surface swept by a line turning about a fixed axis it crosses at a fixed angle; two cones joined at the vertex.
- Generator
- The turning line whose positions make up the cone.
- Nappe
- Either of the two halves of the cone on each side of the vertex.
- Conic section
- The curve in which a plane cuts a double cone.
- Degenerate conic
- A point, a line or a pair of lines, got when the plane passes through the vertex.
- Focus
- A fixed point used in the distance definition of a parabola, ellipse or hyperbola.
- Directrix
- The fixed line of a parabola; every point of the curve is as far from it as from the focus.
- Vertex
- A point where the curve crosses its axis (major or transverse axis for an ellipse or hyperbola).
- Latus rectum
- The chord through a focus perpendicular to the axis (major or transverse axis), ending on the curve.
- Major and minor axes
- For an ellipse, the chord through the foci (length 2a) and the chord perpendicular to it through the centre (length 2b).
- Transverse and conjugate axes
- For a hyperbola, the line through the foci and the line through the centre perpendicular to it; lengths 2a and 2b.
- Eccentricity
- e = c/a, centre-to-focus over centre-to-vertex; below 1 for an ellipse, above 1 for a hyperbola.
- Equilateral hyperbola
- A hyperbola with a = b.
- Locus
- The path traced by a point that moves under a given condition, such as the ellipse traced by a point of a sliding rod.