# Conics Formula Sheet - DOC

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```					                                              Conics Summary
Conic Section                              Standard Form                          Other Info.
Circle                                                                            Derived from the distance
( x  h)  ( y  k )  r
2                    2      2
formula.
Cent re   (h, k )

Parabola - Vertex  (h, k )
p  0 opens up,
Focus (h, k  p )                          ( x  h) 2  4 p ( y  k )
p  0 opens down
Directrix at y  k - p

Foci  ( h  p, k )                         ( y  k ) 2  4 p( x  h)              p  0 opens right,
Directrix at x  h - p                                                            p  0 opens left
 Ellipse - Centre ( h, k )                                                          The longer axis is called the
major axis, the short er axis is
( x  h)   2
( y  k)   2
   Horiz ontal major axis: a > b                                           1       called the minor axis.
a2                 b2                  ‘a’ is the distance from the
Vertices: ( h  a, k )                                                          centre to eac h vertex (the end
Foci:   ( h  c, k )                                                           of the major axis).
( x  h) 2          ( y  k)2             ‘b’ is the distance from the
                  1       centre to the end of the minor
   Vertical major axis: a > b                b2                  a2                  axis.
   ‘c’ is the distance from the
Vertices: (h, k  a)                                                           centre to eac h focus.
Foci: (h, k  c)                                                                 c a b
2      2      2
   Length of major axis = 2a
   Length of minor axis = 2b

   Hyperbola - Centre        (h, k )                                                ‘a’ is the distance from the
centre to eac h vertex.
‘b’ is a point on the conjugate
( x  h) 2         ( y  k )2

   Horiz ontal trans verse axis
(x coefficient is positive)                                             1       axis but is not a point on the
Vertices: (h  a, k )                   a2                 b2                   hyperbola (it helps determine
asymptotes)
Foci:   ( h  c, k )                                                          ‘c’ is the distance from the
centre to eac h focus.
b
Asymptote:       yk          ( x  h)                                                   c a b
2      2      2
a
   N.B. The trans verse axis is
   Vertical trans verse axis              ( y  k)   2
( x  h)   2            not necessarily the longer axis
(y coefficient is positive)                                            1       but is associated with
Vertices: (h, k  a)                    a2                 b2                   whichever variable is positive.

Foci:   (h, k  c)
a
Asymptote : y  k             ( x  h)
b

092                                          Conics Smmary

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