Learning Objectives
- Understand circles, parabolas, ellipses, and hyperbolas
- Learn standard equations and their properties
- Study eccentricity, focus, directrix, and latus rectum
Key Concepts
Circle
Standard form: (x - h)2 + (y - k)2 = r2 (centre (h,k), radius r). General form: x2 + y2 + 2gx + 2fy + c = 0 (centre (-g,-f), radius = √(g2+f2-c), if g2+f2-c > 0). Equation with endpoints of diameter (x₁,y₁) and (x₂,y₂): (x-x₁)(x-x₂) + (y-y₁)(y-y₂) = 0.
Parabola
Standard equation y2 = 4ax: Vertex (0,0), Focus (a,0), Directrix x = -a, Axis: x-axis, Latus rectum = 4a. Other forms: y2 = -4ax (opens left), x2 = 4ay (opens up), x2 = -4ay (opens down). Eccentricity: e = 1 (always for parabola). Parabola is the locus of a point equidistant from focus and directrix.
Ellipse
Standard form (a > b): x2/a2 + y2/b2 = 1. Centre (0,0). Vertices (±a, 0). Foci (±c, 0) where c2 = a2 - b2. Eccentricity: e = c/a < 1. Length of major axis = 2a, minor axis = 2b. Latus rectum = 2b2/a. Sum of distances from any point on ellipse to both foci = 2a.
When a < b: x2/a2 + y2/b2 = 1 with major axis along y-axis: vertices (0, ±b), foci (0, ±c), c2 = b2 - a2.
Hyperbola
Standard form: x2/a2 - y2/b2 = 1. Centre (0,0). Vertices (±a, 0). Foci (±c, 0) where c2 = a2 + b2. Eccentricity: e = c/a > 1. Transverse axis = 2a, Conjugate axis = 2b. Latus rectum = 2b2/a. Asymptotes: y = ±(b/a)x. |Difference| of distances from any point on hyperbola to both foci = 2a.
Conjugate hyperbola: y2/b2 - x2/a2 = 1 (transverse axis along y-axis). Rectangular hyperbola: a = b; equation: x2 - y2 = a2; e = √2; asymptotes are perpendicular.
General Second Degree Equation
ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 represents: Circle (a = b, h = 0), Parabola (h2 = ab), Ellipse (h2 < ab), Hyperbola (h2 > ab).
Summary
Conic sections are curves obtained by intersecting a cone with a plane. Circle (e = 0), Parabola (e = 1), Ellipse (0 < e < 1), Hyperbola (e > 1). Each has standard equations with specific geometric properties: focus, directrix, eccentricity, and latus rectum.
Important Terms
- Eccentricity: Ratio defining the conic shape; e = c/a
- Focus: Fixed point used to define the conic
- Directrix: Fixed line used to define the conic
- Latus rectum: Chord through focus perpendicular to the axis
- Asymptote: Lines that hyperbola approaches but never touches
- Vertex: Point(s) where conic intersects its axis
Quick Revision
- Circle: e = 0; Parabola: e = 1; Ellipse: e < 1; Hyperbola: e > 1
- Parabola y2 = 4ax: focus (a,0), directrix x = -a, LR = 4a
- Ellipse: c2 = a2 - b2; sum of focal distances = 2a
- Hyperbola: c2 = a2 + b2; |difference| of focal distances = 2a
- Ellipse LR = 2b2/a; Hyperbola LR = 2b2/a
- Circle: (x-h)2 + (y-k)2 = r2
- Rectangular hyperbola: a = b, e = √2