|
Examples on Conic Sections
Here are some animations that demonstrate
the changing shape of an ellipse and a hyperbola each with a fixed center
and changing foci and the changing shape of a
parabola as the focus and directrix get closer and closer together. Ellipse,
Hyperbola, Parabola
This
animation shows a conic section transforming from a circle to an ellipse to a
hyperbola as eccentricity changes from 0 to 4 (e = 1 which yields a parabola is
not seen and the ellipses disconnect).
Quicktime Version
Here is a page of
examples and demonstrations illustrating the
formation of a conic section as the intersection of a plane and a cone.
Parabolas
| Example 1
Find an equation of the parabola going
through the points (-2,7), (0,4), (8,12).
|
 |
| Example 2
Find the vertex and focus of the
parabola whose equation is given. Find the equation of its
directrix. Find the endpoints of its latus rectum. Find the
length of the parabolic arc intercepted by the latus rectum.
|
The graph below has the same scale on
each axis.
|
| Example 3
Maple Worksheet
Find an equation of the parabola whose
focus is at (-2,3) and whose directrix has equation x = 4.
The vertex must be halfway between the
focus and directrix (the midpoint of the magenta line segment in the
picture) which would be the point (1,3). Thus from
 |
 |
| Example 4--Section 10.1#81
A solar collector for heating water is
constructed with a sheet of stainless steel that is formed into the
shape of a parabola (see pictures and picture in text). The water
will flow through a pipe that is located at the focus of the parabola.
At what distance from the vertex is the pipe?
The 6 ft and 1 ft dimensions shown in
the textbook picture allow us to model a cross section of the solar
collector as a portion of a parabola with endpoints at (-3,1) and (3,1)
and vertex at (0,0). It's focus would be at (0,k). The task
is to find k since that would give the distance from the vertex to the
focus (pipe).
The standard form of the equation of a
parabola that we could use here would be
 |
 |
| Example 5--Section 10.1#87
Radio and Television Reception
In mountainous areas, reception of
radio and television is sometimes poor. Consider an idealized case
where a hill is represented by the graph of the parabola y = x - x2,
a transmitter is located at the point (-1,1), and a receiver is located
on the other side of the hill at point (x0,0). What is
the closest the receiver can be to the hill so that the reception is
unobstructed?
We need to find x0 such
that the line determined by (-1,1) and (x0,0) is tangent to
the graph of the parabola.
 |
 |
Ellipses
| Example 1
Find an equation of the ellipse with
foci at (-2,-3) and (6,-3) and eccentricity 0.8.
Since the center of the ellipse is the
midpoint of the line segment connecting the foci, the center of the
ellipse is at (2,-3). Since the foci are on a line parallel to the
x-axis the standard form of the equation is
 |
 |
| Example 2
Extra Credit
Prove Theorem 10.4, Reflective
Property of an Ellipse, on page 699 in LHE, 8th edition for the special
case pictured at the right using this ellipse and this particular point
on the ellipse.
 |
 |
| Example 3
Find the area of the ellipse whose
equation is given.
|
To derive the
formula we are computing 4 times the area of the region shown below.
 |
| Example 4--Section 10.1#103
Consider a particle traveling
clockwise on the elliptical path indicated below. The particle
leaves the orbit at the point (-8,3) and travels in a straight line
tangent to the ellipse. At what point will the particle cross the
y-axis.
|
|
Hyperbolas
| Example 1
Find the center, vertices, endpoints
of the conjugate axis, foci, and equations of the asymptotes of the
hyperbola whose equation is given.
|
 |
| Example 2
Find an equation of the hyperbola with
vertices at (1,0) and (-5,0) and the endpoints of its conjugate axis at
(-2,-2) and (-2,2).
The vertices indicate that the
hyperbola would have an equation in standard form as given below.
The center will be midway between the
vertices at (-2,0). The distance between the vertices is 2a and
equals 6 so a = 3. The length of the conjugate axis is 2b and
equals 4 so b = 2. Thus the equation of the hyperbola in standard
form is
 |
 |
| Example 3--Section 10.1#118
Navigation
LORAN
(long distance radio navigation) for
aircraft and ships uses synchronized pulses transmitted by widely
separated transmitting stations. These pulses travel at the speed
of light (186,000 miles per second). The difference in the times
of arrival of these pulses at an aircraft or ship is constant on a
hyperbola having the transmitting stations as foci. Assume that
two stations, 300 miles apart, are positioned on the rectangular
coordinate system at (-150,0) and (150,0) and that a ship is traveling
on a path with coordinates (x,75). Find the x-coordinate of the
position of the ship if the time difference between the pulses from the
transmitting stations is 1000 microseconds (0.001 second).
 |
Click here
to view an animation of the ship moving from an x-coordinate of 150 to
an x-coordinate of 50 along with the changing shape of the right branch
of the hyperbola. Small version
Both Branches |
|
