The Hyperbola
The hyperbola is the set of points P in the plane such that the
absolute value of the difference of the distances from P to two fixed points F1
& F2
is a constant.
 F1P
 F2P
 = k
The fixed points are the foci of the of the hyperbola. Using the
image of the stacked cones again, the shape of the parabola is more apparent:
The general equation for the hyperbola is
, when it is in the
standard position (like above). Notice the similarties to the equation of the
ellipse.
The standard equation for the hyperbola is
The key points for the hyperbola in standard position are:
Information for the hyperbola:
Â· Length of
transverse axis is 2a
Â· Length of
Conjugate axis is 2b
Â· Vertices are at
(a,0) & (a,0)
and the covertices are at (0,b) and (0, b)
o Transverse along
y axis then Vertices (0, Â±
a) ,
covertices at ( Â±
b, 0)
Â· The foci are at
(c,v0) & (c,v0)
and are on the transverse axis
o Transverse along
y axis Foci at (0, Â±
c )
Â· The asymptotes
are at ;
Â· a^{ 2} +
b^{ 2} = c^{ 2}
For the ellipse with the transverse axis on the y axis, the
major points are on the y, and x axis coordinates are zero (flipped points)
The equation is
Example:
Given
,
find the loci and the asymptotes:
This is in the form of an ellipse with the transverse axis along
the y axis. The vertices will be at (0,6) and (0,6).the foci are at
Hence the foci are at (0,
) & (0,).
The asymptotes are at
For hyperbolas not centered at the origin:
The standard form of the equation of a hyperbola centered
at (h, k), with the transverse axis parallel to the xaxis and the
conjugate axis parallel to the yaxis, is
The length of the transverse axis parallel to the xaxis is 2a.
The length of the conjugate axis parallel to the yaxis is 2b.
The vertices are V_{ 1} (h  a, k) and V_{ 2} (h + a,
k).
The covertices are (h, k  b) and (h, k + b).
The foci are F_{ 1} (h  c, k) and F_{ 2} (h + c, k),
which are on the transverse axis.
a^{ 2} + b^{ 2} = c^{ 2} 
The standard form of the equation of a hyperbola centered
at (h, k), with the transverse axis parallel to the yaxis and the
conjugate axis parallel to the xaxis, is
The length of the transverse axis parallel to the yaxis is 2a.
The length of the conjugate axis parallel to the xaxis is 2b.
The vertices are V_{ 1} (h, k  a) and V_{ 2} (h, k +
a).
The covertices are (h  b, k) and (h + b, k).
The foci are F_{ 1} (h  c, k) and F_{ 2} (h + c, k),
which are on the transverse axis.
a^{ 2} + b^{ 2} = c^{ 2} 


