# Conics Formula Sheet

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Conics Summary
Conic Section                      Standard Form                            Graphing Implications
Circle                                                                      Centre (h,k)
( x  h)  ( y  k )  r
2                  2     2

Parabola                                                                    Vertex (h,k)
 opens up if a > 0                ( y  k )  a ( x  h) 2                 ‘a’ gives you the stretch
 opens down if a < 0                                                           (not needed in this unit)

( x  h)  a ( y  k ) 2
   opens right if a > 0
   opens left if a < 0

Ellipse                                                                     Centre (h,k)
The longer axis is called the
( x  h)    2
( y  k)   2
                   1
   Horizontal major axis: a > b                                            major axis, the shorter axis is
2                  2             called the minor axis.
a                  b                 ‘a’ is the distance from the
centre to each vertex (the end of
   Vertical major axis: a > b                                              the major axis)
( y  k )2          ( x  h) 2           ‘b’ is the distance from the
                   1   centre to the end of the minor
a2                 b2                axis.
Length of major axis = 2a
Length of minor axis = 2b

Hyperbola
 Horizontal transverse axis                                                Centre (h, k)
( x  h) 2          ( y  k )2
                  1
(x coefficient is positive)
‘a’ is the distance from the
a2                 b2                centre to the hyperbola.
Equation of the Asymptotes:              ‘b’ does not intersect the
b                   hyperbola, but helps determine
y  k   ( x  h)           the asymptotes (and therefore
a                   the curvature of the hyperbola).

N.B. The transverse axis is not
   Vertical transverse axis
( y  k)    2
( x  h)   2         necessarily the longer axis.
(y coefficient is positive)                                       1
a2                 b2
Equation of the Asymptotes:
a
yk        ( x  h)
b

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