Hyperbolas
section 9.3
hyperbola - the set of all points in a plane such that the difference of the
distances from each point to two fixed points, the foci, is constant
P
F
1
F
2
branches - the two sections of each hyperbola
focal radii - the two segments joining each point on a hyperbola to the two focii
center of the hyperbola - midpoint of segment joining its foci
transverse axis - segment joining the two branches of the hyperbola and
passing through its center
conjugate axis - segment passing through the center and perpendicular to
the transverse axis
asymptotes - lines containing the diagonals of the rectangle whose center
is the center of the hyperbola and whose length and width are
the lengths of the transverse and conjugate axes
vertices of the hyperbola - endpoints of the transverse axis
Equations of the Hyperbola (standard form)
2
(x - h)
-
a2
(y - k)2
b2
=1
when the transverse
axis is horizontal
(y - k)2
a2
2
-
(x - h)
b2
=1
when the transverse
axis is vertical
a = distance from center to a vertex (half the transverse axis)
b = half the conjugate axis
(h, k) is the center
c = distance from center to focus
c 2 = a2 + b 2
2a = difference of focal radii
problem #1 - Find the standard form of the equation of the hyperbola with
foci (- 1, 2) and (5, 2) and vertices (0, 2) and (4, 2).
problem #2 - Sketch the hyperbola whose equation is 4x 2 - y 2 = 16.
Include the asymptotes.
problem #3 - Sketch the hyperbola given by:
4x 2 - 3y 2 + 8x + 16 = 0
problem #4 - Find the standard form of the equation of the hyperbola having
vertices (3, - 5) and (3, 1) and having asymptotes y = 2x - 8
and y = - 2x + 4.
* Continued on next page *
section 9.3 (continued)
The eccentricity of a hyperbola is found in the same way as that of the ellipse.
c
e=
a
However, for the hyperbola, the eccentricity will always have a value greater
than 1, since, in the hyperbola, c > a. If the eccentricity is large, the branches
of the hyperbola are nearly flat; if the eccentricity is close to 1, the branches
are pointed.
vertex
eccentricity = e is large
focus
vertex
eccentricity = e
is close to 1
focus
Applications of the Hyperbola
1. navigation(LORAN)
2. location of objects or events (GPS)
3. paths of some comets
General Equations of Conics
The graph of Ax 2 + Cy 2 + Dx + Ey + F = 0
is one of the following:
1. Circle
if A = C.
2. Parabola
if either A or C equals 0.
3. Ellipse
if A and C have the same sign.
4. Hyperbola
if A and C have different signs.
problem #5 - Classify the graph of each equation:
a. 4x 2 - 9x + y - 5 = 0
b. 4x 2 - y 2 + 8x - 6y + 4 = 0
c. 2x 2 + 4y 2 - 4x + 12y = 0
d. 2x 2 + 2y 2 - 8x + 12y + 2 = 0
** We will omit the last two pages of this section relating to
Rotation of Conics.