The Hyperbola

Definition of the Hyperbola (Geometric)
The final conic that we will study is called the hyperbola.
Definition
Let P and Q be two fixed points, and
c be a constant.
Then the set of points in the plane such that
QR  PR = c
is a
hyperbola. 
Let C be the center of the hyperbola and V be a vertex.
We define the eccentricity of a hyperbola as
Notice that this is the same definition as the definition for the eccentricity
of the ellipse. Notice also that the eccentricity of a hyperbola is
always greater than 1. We call the two pieces of the hyperbola the
branches of the hyperbola. The line segment that connects the
vertices is called the transverse axis, while the other axis is called
the conjugate axis.

The Equation of the Hyperbola
The standard equation of the hyperbola with center at (h,k)
is
The
Standard Equation of a Hyperbola
(x  h)^{2}/a^{2} 
(y 
k)^{2}/b^{2} = 1 
When the center is at the origin the equation becomes
The Hyperbola With Center at the
Origin
x^{2}/a^{2}  y^{2}/b^{2} = 1 
Solving for y we get:
y^{2}/b^{2} = x^{2}/a^{2}  1
which reduces to
y^{2} = x^{2}b^{2}/a^{2} 
b^{2}
or
y = + ( x^{2}b^{2}/a^{2}  b^{2})^{1/2}
If x is large the b^{2} term goes away, hence we get the equation
y = + b/a x
y = b/a x
and y = b/a x
are the asymptotes of the hyperbola.
Example:
Sketch the hyperbola
(x  2)^{2}/4  (y + 1)^{2}/9 =
1
Solution:
We see that the center is at (2,1) , the vertices are at
(0,1) and (4,1),
and the asymptotes have slope y = 3/2 and 3/2.

Identifying Conics:
We can tell which type of conic an equation represents by the signs in front
of the square term, the number of square terms and the coefficients in front
of the square terms.

If there is only one square term then the conic is a
parabola

If the signs differ then the conic is a hyperbola

If the signs and the coefficients are the same, then the conic is a
circle

If the signs are the same but the coefficients differ, then the conic is
an ellipse.
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