Conic Sections and Coordinate Geometry
Standard forms
Parabola y^2=4ax: focus (a,0), directrix x=-a. Ellipse x^2/a^2+y^2/b^2=1 (a>b): c^2=a^2-b^2, foci (±c,0), eccentricity e=c/a<1. Hyperbola x^2/a^2-y^2/b^2=1: c^2=a^2+b^2, foci (±c,0), e=c/a>1. Rectangular hyperbola xy=c^2 has parametric form (ct,c/t). For shifted conics, replace x by x-h and y by y-k before reading off the centre. Eccentricity measures shape: e=0 circle, 0<e<1 ellipse, e=1 parabola, e>1 hyperbola.
Parametric and tangents
Parabola (at^2,2at); tangent ty=x+at^2. Ellipse (a cosθ,b sinθ); hyperbola (a secθ,b tanθ). For a general second-degree equation, completing the square identifies the conic; the discriminant B^2-4AC classifies it.
Worked example
For y^2=8x, 4a=8 so a=2. At t=1 the point is (2,4). Tangent: ty=x+at^2 gives y=x+2. Check: substituting y=x+2 into y^2=8x gives (x+2)^2=8x, i.e. x^2-4x+4=0=(x-2)^2, a repeated root, so the line is tangent.
Traps
Mixing c^2=a^2-b^2 with c^2=a^2+b^2; using b instead of a for eccentricity; wrong directrix sign; forgetting that parametric t can be any real but θ is periodic; tangents at points not on the curve; not checking whether a denominator is zero; losing the negative branch.