Conics
Hyperbola
Conics
Hyperbola Cross Section
Hyperbola
A hyperbola is the set of all points in a plane
whose distances from two fixed points in the
plane have a constant difference. The fixed
points are the foci of the hyperbola. The line
through the foci is the focal (major) axis. The
point on the focal axis midway between the
foci is the center. The points where the
hyperbola intersects its focal axis are the
vertices of the hyperbola.
Horizontal
Vertical
Axis
The chord lying on the focal axis connecting the
vertices is the transverse axis.
Semi transverse axis—the distance from the
center to the vertex—(a)
Hyperbola with center (0,0)
Horizontal
Vertical
STD
Focal Axis
Foci
Vertices
Semi-Tran Axis
Semi-Conjugate Axis
Pythagorean Rel.
Asymptotes
x-axis
(±c,0)
(±a,0)
a
b
c²=a²+b²
y=±(b/a)x
y-axis
(0, ±c)
(0, ±a)
a
b
c²=a²+b²
y=±(a/b)x
E1
• 1) Plot
x² - y² = 1
E1
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.
E2
Plot
(x²/9) – (y²/16) = 1
E2
• 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.
Hyperbola with Center (h,k)
E3
Plot
(y²/49) – (x²/4) = 1
E3
•
•
•
•
•
•
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.