The hyperbola is created
when the intersecting plane is not parallel to an edge of the
cone. The following diagram illustrates this intersection, the
hyperbola. Although the diagram shows the plane parallel to the axis
of the cone, in general this does not have to be the case
Notice there are two
intersections, one with each
nappe of the cone. These intersections
are called branches of the hyperbola. Now imagine slicing off the nappes with the plane and rotating it toward us as shown to get the
next diagram.
Notice that the curves do not
touch the edges of the cone. We say that these curves are
asymptotic
(ă-sĭm-tŏt’-ĭk)
to the edges of the cone. As these curves spread, so doe the edges of
the cone. The curves will get closer and closer to the edges of the
cone but will never touch it. These lines along the edge of the cone
are the asymptotes (ă’-sĭm-tōts)
of the hyperbola.
The
next diagram shows this cone rotated -900
and
the XY axis superimposed onto the intersection plane.
On
this diagram the line determined by the two
foci
(fō’-sē,
plural of focus),
f1
and f2
intersects the hyperbola in two places, the vertices
(plural of vertex).
The line segment between the foci is called the
transverse
axis. The center
of the hyperbola lies on the transverse axis at a distance 'a' from
either vertex. In this diagram the center is at the origin.
Now,
let's define the properties of the hyperbola.
Note
above, that a rectangle is formed by the vertical lines that
intersect each vertex and the asymptotes, then horizontally from
these intersections from one asymptote to the other. We will label
the distance from the origin to these vertical lines
a,
and the distance from the origin to horizontal lines of the
rectangle b.
These
distances, a and b, are called the x radius and
y radius
respectively of the hyperbola.
After labeling the distance
from the origin to the corner of the box c,
we use Pythagorean's Theorem to relate a, b, and c, as shown. c
is the distance from the origin to either focal point. The blue arrow
shows the effect of rotating the the blue line segment about the
origin down to the X axis. The distance to the focus is c.
The asymptotes are lines that intersect at the center
of the hyperbola. When the center is the origin then the equations
for each have no intercept, so they will be in the form y = mx.
Notice the the slope of each asymptote is the negative
of the slope of the other asymptote.
The next diagram shows a
point P on the right branch of the hyperbola. D1
is the distance from focus f1 to P and d2
is the distance from f2 to P.
The definition of the
hyperbola:
1) the distance between the foci f1
and f2 is always greater than 2a and
2) all points P whose distances to f1
and f2 subtracted is 2a.
| d1 - d2
| = 2a
The following is a hyperbola whose center is at
(h,k)
This is the general equation for a hyperbola
If we interchange the signs on the terms on the left
hand side of the equation, we rotate the hyperbola 900.
The next diagram illustrates this effect. The center
of the hyperbola is the origin. All relationships still hold.
Notice that the constants stay with the variables.
Examples.
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1) Plot
a2 = 1, so a = 1
b2 = 1, so b = 1
This equation is for a hyperbola
whose center is at the origin. So sketch in the green square. Draw
the green lines through the diagonals of the square, these are the
asymptotes. The vertices occur at y=0, substituting into the
equation we get: x2 - 0 = 1. x = ±
1
Plot the vertices (red dots) and sketch
the branches without crossing the asymptotes.
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2) Plot
Here a2 = 9, so a = 3
and b2 = 16, so b = 4
when y = 0, x2 = 9 so
the vertices are
at x = ±
3
Plot the green rectangle,
sketch in the asymptotes, and mark the vertices. Now sketch in the
hyperbola without crossing the asymptotes.
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3) Plot
Notice that the signs have
interchanged, the minus is in front of x2 and the plus
sign is in front of y2. This is a hyperbola the opens
along the Y axis.
We have b2 = 49, so b =
7
and a2 = 4, so a = 2.
The vertices are at x=0,
substituting in we get y2 / 49 - 0 = 1
which is y2 = 49 so y =
±7
Plot the green rectangle, the
asymptotes through its diagonals and the vertices then sketch in
the hyperbola without crossing the asymptotes.
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4) Plot
This is a translated hyperbola
with
h = 3 and k = -2. The center of
this hyperbola is at (h,k) = (3, -2).
a2 = 16, so a = 4
b2 = 36, so b = 6
Plot this green rectangle about
the center (3, -2) as shown.
Sketch in the asymptotes through
the diagonals of this rectangle.
The x term has the plus sign so
this hyperbola opens along the X axis. So the vertices will be on
the dotted red line (y = -2 ). Substituting in this value for y
and solving for x we have:
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The foci are shown.
Recall how a, b, and c are
related:
The foci are relative to the center, not the origin. 3 + 7.2
and 3 - 7.2 are the x values.
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5a) Plot
This is a special hyperbola. It
represents the inverse relationship between x and y. This equation
can be written
The X and the Y axes are the
asymptotes.
Each branch alone is symmetric
about the line y = x (red dashed line).
Point P has a symmetric point P1
across this line.
The branches are mirror images of
each other across the the blue dashed line
y = -x. Each point P has its mirror image on the other branch
P2.
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5b) Plot
This can be written
The difference between this equation and the previous equation
in example 5a is the negative sign. The effect is to rotate the
hyperbola into the other two quadrants. Notice the lines of
symmetry and reflection as discussed in 5a.
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