
A conic section is the intersection of a plane with a cone. can be defined: as the locus of all points P, such that the ratio of the distance from P to a fixed point (the focus), to the distance from P to a fixed line (the directrix), is constant. The value of this constant is known as the eccentricity, e. The eccentricity can also be interpreted as the fraction of the distance from the center to a vertex along the semimajor axis at which the focus lies.
The general form of the equation of a conic section is ax² + 2hxy + by² + 2gx + 2fy + c = 0
If h²<ab, the conic is an ellipse; the plane intersects just one side of the cone at a greater angle than the cone's central angle.
If h²>ab, the conic is a hyperbola; the plane intersects both sides of the cone at a smaller angle than the cone's central angle.
If h²=ab, the conic is a parabola; the plane intersects just one side of the cone at exactly the same angle as the cone's central angle.
In order to eliminate the xy term from the equation
ax² + 2hxy + by² + 2gx + 2fy + c = 0,
the axes should be rotated counterclockwise through an angle equal to
½ arctan( 2h/(ab) )
Hyperbola parameters
2a is the length of the transverse axis, which connects the vertices; 
A hyperbola that opens left and right is (xh)²/a²  (yk)²/b² = 1 The vertices are at (ha,k) and (h+a,k)
The foci are at (hae,k) and (h+ae,k), 
A hyperbola that opens up and down is (yk)²/b²  (xh)²/a² = 1 The vertices are at (h,ka) and (h,k+a)
The foci are at (h,kae) and (h,k+ae), 
A "rectangular hyperbola" is xy=k
Parabola parameters
The eccentricity, e, of a parabola is defined as 1. 
A parabola with a vertical axis of symmetry is The axis of symmetry of the parabola is x = b/(2a). The vertex is
If the equation of the parabola is rewritten (yk) = a(xh)² then the vertex is (h,k), the focus is (h,k+1/(4a)) and the directrix is y=k1/(4a). The latus rectum is the chord of the parabola parallel to the
directrix that passes through its focus, 
A parabola with its axis parallel to the x axis is The axis of symmetry of the parabola is y = b/(2a). The vertex is
If the equation of the parabola is rewritten (xh) = a(yk)² then the vertex is (h,k), the focus is (h+1/(4a),k) and the directrix is x=h1/(4a). The latus rectum is the chord of the parabola parallel to the directrix that passes through its focus, x=h+1/(4a). The length of the latus rectum is 1/a. 
Ellipse parameters
2a is the length of the major axis, which is the longer axis; 
An ellipse with a horizontal major axis is
The foci are at (hae,k) and (h+ae,k), 
An ellipse with a vertical major axis is
The foci are at (h,kae) and (h,k+ae), 
Mathwords: foci of an ellipse, focus of a parabola, foci of a hyperbola
Planet Math: conic section, and the Dandelin sphere
Wikipedia: Rotation of Axes
Mudd Math Fun Facts: eccentricity of conic sections
Mathworld: eccentricity  can be interpreted as the fraction of the distance along the semimajor axis at which the focus lies.
Cyclic quadrilaterals  bunch of theorems related to inscribing polygons in conic sections
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