Conic section

Circle

Circle[point M, number r]

Circle with midpoint M and radius r

Circle[point M, segment s]

Circle with midpoint M and radius = Length[s]

Circle[point M, point A]

Circle with midpoint M through A

Circle[point A, point B, point C]

Circle through A, B and C

Ellipse

Ellipse[point F, point G, number a]

Ellipse with foci F, G and principal axis length a. Condition: 2a > Distance[F,G]

Ellipse[point F, point G, segment s]

Ellipse with foci F, G and principal axis length a = Length[s]

Hyperbola

Hyperbola[point F, point G, number a]

Hyperbola with foci F, G and principal axis length a. Condition: 0 < 2a < Distance[F,G]

Hyperbola[point F, point G, segment s]

Hyperbola with foci F, G and principal axis length a = Length[s]

Parabola

Parabola[point F, line g]

Parabola with focus F and directrix g

Conic

Conic[point A, point B, point C, point D, point E]

Conic section through five points (no four lie on one line)