The activities in this lesson are not graded. However, I strongly advised you to complete them to prepare for the quiz ANGEL.
When a plane intersects two right circular cones at the same time, the conic section formed is a hyperbola.
Hyperbola
The Geometric Definition of the Hyperbola
A hyperbola is the set of all points in a plane, the difference of whose distances from two fixed points is a positive constant. The two fixed points, F_{1} and F_{2} , are called the foci.The midpoint of the line segment joining the foci is called the center of the hypebola.

Notice that the geometric definition above is very similar to the geometric definition of the ellipse. Recall that for a point to lie on the graph of an ellipse, the sum of the distances from the point to the two foci is constant. For a point to lie on the graph of a hyperbola, the difference between the distances from the point to the two foci is constant. Because subtraction is not commutative, we will consider the absolute va lue of the difference in the distances between a point on the hyperbola and the foci to ensure that the constant is positive. Thus for any two points P and Q that lie on the graph of a hyperbola
For any points P and Q that lie on the graph of a hyperbola,
The graph of a hyperbola has two branches. Each branch looks somewhat like a parabola but are certainly not parabolas because the branches do not satisfy the geometric definition of the parabola. Every hyperbola has a center, two vertices and two foci.
The vertices are located at the endpoints of an invisible line segment called the focal axis or transverse axis. The transverse axis is either parallel to the xaxis (horizontal transverse axis) or it is parallel to the yaxis (vertical transverse axis).
The center of a hyperbola is located midway between the two vertices (or two foci). The hyperbola has another invisible line segment called the conjugate axis which passes through the center and lies perpendicular to the transverse axis.
Each branch of the hyperbola approaches (but never intersects) a pair of lines called asymptotes. These lines intersect at the center of the hyperbola and have the property that as a point P moves along the hyperbola away from the center, the distance between P and one of the asymptotes approaches zero.
A reference rectangle is typically used as a guide to help sketch the asymptotes. The reference rectangle is an imaginary rectangle whose midpoints of each side are the vertices of the hyperbola or the endpoints of the conjugate axis. The asymptotes pass diagonally through opposite corners of the reference rectangle.
Focal axis or Transverse axis:
Parallel to the xaxis if the hyperbola is horizontal
Parallel to the yaxis if the hyperbola is vertical
Conjugate axis: Passes through the center and is perpendicular to the transverse axis.
Vertices: Located at the endpoints of the transverse axis
Center: Midpoint of the segment joining the two vertices and the line segment joining the two foci.
Reference Rectangle : Rectangle whose midpoints of each side are the vertices of the hyperbola or the endpoints of the conjugate axis.
Asymptotes: Pass diagonally through opposite corners of the reference rectangle. Each branch of the hyperbola approaches (but never intersects) the asymptotes
(a) Hyperbola with a Horizontal Transverse axis
(b) Hyperbola with a Vertical Transverse Axis
It is traditional in the study of hyperbolas to denote the distance between the vertices by 2a, the distance between the foci by 2c (see the fiqure below) and to define the quantity b as
To derive the equation of a hyperbola, consider a hyperbola with a horizontal transverse axis and following the below Geometric Properties of a Hyperbola Activity
By the geometric definition of a hyperbola we know that for any point P(x,y) that lies on the hyperbola, the difference of the distances from P to the two foci is a constant. From the Geometric Properties of a Hyperbola Activity, it shown that this constant is equal to 2a, the length of the transverse axis.
Hyperbola has the follow properties
Since , we use the distance formula and abtain
then square both sides and simplifying many more steps (I won't go through the detail) yields
The equation of a hyperbola is simplest if the coordinate axes are posititioned so that the center of the hyperbola is at the orgin and the foci are on the xaxis or yaxis.
Figure1
REMARK: There is a trick that can be used to avoid memorizing the equations of the asymptotes of a hyperbola. They can be obtained, when needed, by substituting 0 for the 1 on the right side of the hyperbola equation, and then solving for y in terms of x. For example, for the hyperbola
we would write
or
take the square root
which are the equations for the asymptotes.
Hyperbolas can be graphed from their standard equations using four basic steps:
Rough Sketch
Horizontal Transverse Axis (Parallel to the xaxis)
Center: (h,k) Vertices: (ha,k) and (h+a,k) Foci: (hc,k) and (h+c,k) Asymptotes: 
Vertical Transverse Axis (Parallel to the yaxis)
Center: (h,k) Vertices: (h,k+a) and (h,ka) Foci: (h,kc) and (h,k+c) Asymptotes:

A hyperbola such as in the last example is called equilateral if a = b.
Hyperbolas have many applications. Some comets are only seen once in our solar system as they travel through the solar system on the path of a hyperbola with the sun at a focus. On October 14, 1947, Chuck Yeager became the first person to break the sound barrier. As an airplane moves faster than the speed of sound a cone shaped shock wave is produced. The cone intersects the ground in the shape of a hyperbola.
When two rocks are simultaneously tossed into a calm pool of water, ripples move outward in the form of concentric circles. These circles intersect in points that form a hyperbola.
Another application of hyperbolas is in the longrange navigation system LORAN. This system uses transmitting stations in three locations to send out simultaneous signals to a ship or aircraft. The difference in the arrival times of the signals from one pair of transmitters is recorded on the ship or aircraft. This difference is also recorded for signals from another pair of transmitters. For each pair, a computation is performed to determine the difference in the distances from each member of the pair to the ship or aircraft. If each pair of differences is kept constant, two hyperbolas can be drawn. Each has one of the pairs of transmitters as foci, and the ship or aircraft lies on the intersection of two of their branches. See diagram below.
Quiz your concepts and vocabulary
Other Mathematics Resources: www.learner.org
Algebra: In Simplest Terms, look for video:
Ellipse and Hyperbola
Equation 
Type of Conic Section 
Graph 
Only one variable is squared, so this cannot be a cirlce, an ellipse, or a hyperbola. Find an equivalent equation by minus 4y, add 4 to both sides of the equation and then factor: This is an equation of a parabola 

Both variables are squared, so this cannot be a parabola. The squared terms are added, so this cannot be a hyperbola. Divide by 3 on both sides to find an equivalent equation: This is an equation of a circle. 

Both variables are squared, so this cannot be a parabola. Add 4x^{2} on both sides to find an equivalent equation:
The squared terms are added, so this cannot be a hyperbola. The coeffiecient of x^{2} and y^{2} are not the same, so this is not a circle. Divided by 16 on both sides to find an equivalent equation: This is an equation of an ellipse. 

Both variables are squared, so this cannot be a parabola. Subtract 4y^{2} on both sides to find an equivalent equation: The square terms are not added, so this cannot be a circle or an ellipse. Divided by 36 on both sides to find an equivalent equation: This is an equation of a hyperbola.

Review the conis section.