1. Tracking particles in particle physics 2. Gas properties including Boyle's Law, Charles' Law, Ideal Gas Law, etc. 3. designing lens 4. Analyzing capillary forces 5. The math of rainbows

6. Orbits of some space craft

--actually they are used in real life. parabolas are seen in "parabolic microphones" or satellites. and there are others for both ellipses and hyperbolas.

Hyperbolas may be seen in many sundials. On any given day, the sun revolves in a circle on the celestial

sphere, and its rays striking the point on a sundial traces out a cone of light. The intersection of this cone

with the horizontal plane of the ground forms a conic section, by definition. At most populated latitudes

and at most times of the year, this conic section is a hyperbola. In practical terms, the shadow of the tip of

a pole traces out a hyperbola on the ground over the course of a day. The shape of this hyperbola varies

with the geographical latitude and with the time of the year, since those factors affect the cone of the

sun's rays relative to the horizon. The collection of such hyperbolas for a whole year at a given location

was called a pelekinon by the Greeks, since it resembles a double-bladed axe.

A hyperbola is the basis for solving trilateration problems, the task of locating a point from the differences

in its distances to given points — or, equivalently, the difference in arrival times of synchronized signals

between the point and the given points. Such problems are important in navigation, particularly on water;

a ship can locate its position from the difference in arrival times of signals from

a LORAN or GPS transmitters. Conversely, a homing beacon or any transmitter can be located by

comparing the arrival times of its signals at two separate receiving stations; such techniques may be used

to track objects and people. In particular, the set of possible positions of a point that has a distance

difference of 2a from two given points is a hyperbola of vertex separation 2a whose foci are the two given

points.

The paths followed by any particle in the classical Kepler problem is a conic section. In particular, if the

total energy E of the particle is greater than zero (i.e., if the particle is unbound), the path of such a

particle is a hyperbola. This property is useful in studying atomic and sub-atomic forces by scattering

high-energy particles; for example, the Rutherford experiment demonstrated the existence of an atomic

nucleus by examining the scattering of alpha particles from gold atoms. If the short-range nuclear

interactions are ignored, the atomic nucleus and the alpha particle interact only by a repulsive Coulomb

force, which satisfies the inverse square law requirement for a Kepler problem.

As shown first by Apollonius of Perga, a hyperbola can be used to trisect any angle, a intensely studied

problem of geometry. Given an angle, one first draws a circle centered on its middle point O, which

intersects the legs of the angle at points A and B. One next draws the line throughA and B and constructs

a hyperbola of eccentricity ε=2 with that line as its transverse axis and B as one focus. The directrix of the

hyperbola is the bisector of AB, and for any point P on the hyperbola, the angle ABP is twice as large as

the angle BAP. Let P be a point on the circle. By the inscribed angle theorem, the corresponding center

angles are likewise related by a factor of two, AOP = 2×POB. But AOP+POB equals the original angle

AOB. Therefore, the angle has been trisected, since 3×POB = AOB.

Real life application of line, parabola, hyperbola and exponential graphs.

Relate a real life application to the specific graph(s) line, parabola, hyperbola, exponential. 

Then describe the characteristics of each application as related to the graph. 

Real life expressions  - We encounter rational expressions all the time in the real world, often without even realizing it. Now it's your turn to come up with a real-world example from your work or personal life o ...

Geometry in Real-Life  - For geometry I am having trouble to come up with an example for the question below that illustrate numbers. I need an example with numbers and some geometry calculations so that we can clearly see ...

Think of a real life example of a system of linear equations.  - Think of a real life example of a system of linear equations.

Real-Life Applications of Functions  - Describe a real world situation that could be modeled by a function that is increasing, then constant, then decreasing. What would be the difficulties associated with modeling this situation?

All of the graphs occur in real life. Find a real-life application of each graph.  - All of the graphs occur in real life. Find a real-life application of each graph. Relate the application to the specific graph (line, parabola, hyperbola, exponential). Describe the characteristic ...

Real-life situation represented by a exponential function  - Think of a real-life situation that can be represented by a exponential function; translate the situation to the function; solve the function and represent it graphically. I want to use my examp ...

Exponential and Logarithmic Functions : Real-Life Applications  - 1. Explain the relevance and application of exponential functions in real-life situations. 2. Explain the relevance and application of logarithmic functions in real-life situations. 3. Think o ...

How do I create a real-life situation that fits into the equation(x+3) (x-4) = 0 and express the situation as the same equation? - How do I create a real-life situation that fits into the equation(x+3) (x-4) = 0 and express the situation as the same equation?

Equation - Creating a real-life situation that fits the equation (x + 4)(x - 7) = 0 how would I express the situation as the same equation? - Creating a real-life situation that fits the equation (x + 4)(x - 7) = 0 how would I express the situation as the same equation?

Explain the relevance and application of exponential functions in real-life situations  - Explain the relevance and application of exponential functions in real-life situations

Graphing to show data  - Please give a real-life example that uses graphing to show data.

Radical Expressions in Real Life  - Consider how you might apply radical expressions to your daily life. Explain this application, and discuss what the equation might be.

The postal service provides at least one good example of a function with jump discontinuities. What is one such function, taken from the postal service or some other real life example? (Explain why there are jump discontinuities.)   - The postal service provides at least one good example of a function with jump discontinuities. What is one such function, taken from the postal service or some other real life example? (Explain why th ...

Real-life word problem  - Provide a word problem and solution dealing with everyday life. Please provide your own real-world example from your work or life using the slope and y-intercept.

What are the slope and y-intercept in your example?   - Please provide your own real-world example from your work or life using the slope and y-intercept. What are the slope and y-intercept in your example?

Radical Expressions in Real-life  - How would you apply radical expressions in everyday life? Give an example of the expression.

All of the graphs occur in real life. Find a real-life application of each graph. Relate the application to the specific graph (line, parabola, hyperbola, exponential). Describe the characteristics of each application as related to the graph.   - All of the graphs occur in real life. Find a real-life application of each graph. Relate the application to the specific graph (line, parabola, hyperbola, exponential). Describe the characteristic ...

Real-life fractions  - What are some examples of real-life situations where fraction operations (+,-,*,/) might be used? Select one, and discuss your methodology for solving problems in this situation.

Create ways to teach surface area and volume to students. (Real-Life Example)  - Creating ways to teach surface area and volume. I am especially interested in methods which will help students connect geometry to life in the real world.

Lograithmic function and application  - Think of a real-life situation that can be represented by a logarithmic function, translate the situation to the function, and solve the function and represent it graphically.

Real-Life Application of a Linear Equation  - Linear equations provide information on how quickly data is rising or falling, known as the slope of the equation. Find a real-world application of a linear equation and discuss the meaning of the equ ...

Real-Life Applications of Ratios and Proportions   - Describe an application for the use of ratios or proportions and how it could be useful in your daily life?

Rational expressions and their real-life applications  - Based on the eduation and learning curve of rational expressions and their applications, consider how someone might apply rational expressions in their daily life. Explain this application, and discu ...

Real-Life Applications of Functions Versus Relations and Reversal of Variables  - In the real world, what might be a situation where it is preferable for the data to form a relation but not a function? If the variables in an equation were reversed, what would happen to the graph ...

Real Life Applications of Linear Functions  - Using the Library, web resources, and/or other materials, find a real-life application of a linear function. State the application, give the equation of the linear function, and state what the x and y ...

The Definition of a Hyperbola

A hyperbola is the strangest-looking shape in this section. It looks sort of like two back-to-back

parabolas. However, those shapes are not exactly parabolas, and the differences are very important.

Surprisingly, the definition and formula for a hyperbola are very similar to those of an ellipse.

Definition of a Hyperbola

Take two points. (Each one is a focus; together, they are the foci.) A hyperbola is the set of all points

in a plane that have the following property: the distance from the point to one focus, minus the

distance from the point to the other focus, is some constant.

The entire definition is identical to the definition of an ellipse, with one critical change: the word “plus”

has been changed to “minus.”

One use of hyperbolas comes directly from this definition. Suppose two people hear the same noise,

but one hears it ten seconds earlier than the first one. This is roughly enough time for sound to travel 2

miles. So where did the sound originate? Somewhere 2 miles closer to the first observer than the

second. This places it somewhere on a hyperbola: the set of all points such that the distance to the

second point, minus the distance to the first, is 2.

Another use is astronomical. Suppose a comet is zooming from outer space into our solar system,

passing near (but not colliding with) the sun. What path will the comet make? The answer turns out to

depend on the comet’s speed.

If the comet’s speed is low, it will be trapped by the sun’s gravitational pull. The resulting shape will be an elliptical orbit.

If the comet’s speed is high, it will escape the sun’s gravitational pull. The resulting shape will be half a hyperbola.

TABLE 1

We see in this real life example, as in the definitions, a connection between ellipses and hyperbolas.

The Formula of an Hyperbola

With hyperbolas, just as with ellipses, it is crucial to start by

distinguishing horizontal from vertical. It is also useful to pay close attention to which aspects are

the same as ellipses, and which are different.

Horizontal Vertical

x2

a2

−

y2

y2

a2

−

x2

Horizontal Vertical

b2

=1

b2

=1

TABLE 2: Mathematical Formula for a Hyperbola with its Center at the Origin

And of course, the usual rules of permutations apply. For instance, if we replace x with x–h, the

hyperbola moves to the right by h. So we have the more general form:

Horizontal Vertical

(x−h)2

a2

−

(y−k)2

b2

=1

(y−k)2

a2

−

(x−h)2

b2

=1

TABLE 3: Formula for a Hyperbola with its Center at xxx(h,k)

The key to understanding hyperbolas is understanding the three constants a, b, and c.

Horizontal Hyperbola Vertical Hyperbola

Where are the foci? Horizontally around the center Vertically around the center

How far are the foci from the center?

c c

What is the “transverse The (horizontal) line from one vertex The (vertical) line from one vertex

Horizontal Hyperbola Vertical Hyperbola

axis”? to the other to the other

How long is the transverse axis?

2a 2a

Which is biggest? c is biggest. c>a, and c>b. c is biggest. c>a, and c>b.

crucial relationship c2=a2+b2 c2=a2+b2

TABLE 4

Having trouble keeping it all straight? Let’s make a list of similarities and differences.

Similarities between Hyperbolas and Ellipses

The formula is identical, except for the replacement of a+ with a-.

The definition of a is very similar. In a horizontal ellipse, you move horizontally a from the center

to the edges of the ellipse. (This defines the major axis.) In a horizontal hyperbola, you move

horizontally a from the center to the vertices of the hyperbola. (This defines the transverse

axis.)

b defines a different, perpendicular axis.

The definition of c is identical: the distance from center to focus.

Differences Between Hyperbolas and Ellipses

The biggest difference is that for an ellipse, a is always the biggest of the three variables; for a

hyperbola, c is always the biggest. This should be evident from looking at the drawings (the foci

are inside an ellipse, outside a hyperbola). However, this difference leads to several other key

distinctions.

For ellipses, a2=b2+c2. For hyperbolas, c2=a2+b2.

For ellipses, you tell whether it is horizontal or vertical by looking at which denominator is

greater, since a must always be bigger than b. For hyperbolas, you tell whether it is horizontal

or vertical by looking at which variable has a positive sign, the x2 or they2. The relative sizes

of a and b do not distinguish horizontal from vertical.

In the example below, note that the process of getting the equation in standard form is identical

with hyperbolas and ellipses. The extra last step—rewriting a multiplication by 4 as a division by

1

4

 —can come up with ellipses as easily as with hyperbolas. However, it did not come up in the last

example, so it is worth taking note of here.

EXAMPLE 1: Putting a Hyperbola in Standard Form

Graph3x2–12y2–18x–24y+12=0

The problem. We recognize this as a hyperbola because it has an x2 and a y2term, and have different signs (one is positive and one negative).

3x2–18x−12y2–24y=-12

Group together the x terms and the y terms, with the number on the other side.

3

x2–6x

–12

y2+2y

=-12

Factor out the coefficients of the squared terms. In the case of the y2 for this particular equation, the coefficient is minus 12.

3

x2–6x+9

Complete the square twice. Adding 9 inside the first parentheses adds 27; adding 1 inside the second set subtracts 12.

–12

y2+2y+1

=-12+27–12

3(x−3)2–12(y+1)2=3

Rewrite and simplify.

(x−3)2–4(y+1)2=1Divide by 3, to get a 1 on the right. Note, however, that we are still not in standard form, because of the 4 that is multiplied by (y+1)2. The standard form has numbers in the denominator, but not in the numerator.

(x−3)2−

(y+1)2

1

4

=1

Dividing by

1

4

is the same as multiplying by 4, so this is still the same equation. But now we are in standard form, since the number is on the bottom.

TABLE 5

However, the process of graphing a hyperbola is quite different from the process of graphing an

ellipse. Even here, however, some similarities lurk beneath the surface.

EXAMPLE 2: Graphing a Hyperbola in Standard Form

Graph (x−3)2–

(y+1)2

1

4

=1

The problem, carried over from the example above, now in standard form.

Center: (3,–1)Comes straight out of the equation, both signs changed, just as with circles and ellipses.

a=1 b= The square roots of the denominators, just as with the ellipse. But how

1

2

do we tell which is which? In the case of a hyperbola, the a always goes with thepositive term. In this case, the x2 term is positive, so the term under it is a2.

Horizontal hyperbolaAgain, this is because the x2 term is positive. If the y2 were the positive term, the hyperbola would be vertical, and the number under the y2 term would be considered a2.

c=\ 12+ (

1

2

)2 =\

5

4

=

\ 5

2

Remember that the relationship is different: for hyperbolas, c2=a2+b2

Now we begin drawing. Begin by drawing the center at (3,–1). Now, since this is a horizontal ellipse, the vertices will be aligned horizontally around the center. Since a=1, move 1 to the left and 1 to the right, and draw the vertices there.

In the other direction—vertical, in this case—we have something called the “conjugate axis.” Move up and down by b (

1

2

in this case) to draw the endpoints of the conjugate axis. Although not part of the hyperbola, they will help us draw it.

Draw a box that goes through the vertices and the endpoints of the conjugate axis. The box is drawn in dotted lines to show that it is not the hyperbola.

Draw diagonal lines through the corners of the box—also dotted, because they are also not the hyperbola.These lines are called the asymptotes, and they will guide you in drawing the hyperbola. The further it gets from the vertices, the closer the hyperbola gets to the asymptotes. However, it never crosses them.

Now, at last, we are ready to draw the hyperbola. Beginning at the vertices, approach—but do not cross!—the asymptotes. So you see that the asymptotes guide us in setting the width of the hyperbola, performing a similar function to the latus rectum in parabolas.

TABLE 6

The hyperbola is the most complicated shape we deal with in this course, with a lot of steps to

memorize.

But there is also a very important concept hidden in all that, and that is the concept of

an asymptote. Many functions have asymptotes, which you will explore in far greater depth in more

advanced courses. An asymptote is a line that a function approaches, but never quite reaches. The

asymptotes are the easiest way to confirm that a hyperbola is not actually two back-to-back

parabolas. Although one side of a hyperbola resembles a parabola superficially, parabolas do not have

asymptotic behavior—the shape is different.

Remember our comet? It flew into the solar system at a high speed, whipped around the sun, and flew

away in a hyperbolic orbit. As the comet gets farther away, the sun’s influence becomes less

important, and the comet gets closer to its “natural” path—a straight line. In fact, that straight line is

the asymptote of the hyperbolic path.

Before we leave hyperbolas, I want to briefly mention a much simpler equation: y=

1

x

 . This is the equation of a diagonal hyperbola. The asymptotes are the x and y axes.

Figure 1: y=

1Although the equation looks completely different, the shape is identical to the hyperbolas we have

been studying, except that it is rotated 45°.

"Application of hyperbola" Questions & Answers

From   Yahoo Answers

Question : I am looking for a real life application of a hyberbola for an alegbra class.

Answer : Picture a lamp close to a wall. The lamp has a shade that is a cylinder, (shaped like a tin can with the top and bottom cut off). The shade is parallel to the wall. When the light is on, the shape of the light on the wall is a hyperbola. Really!

Question : Math project need answers fast plz?

Answer : Parabola's real world application include satellite dishes, which engineers take into account into their design for data reception. Hyperbola's example are Focus lenses designs which will determine focal points depending on your curve design. And Ellipses probably race track desings that help designers take into account top speeds and such depending on shape.

Question : I have a math final coming up, and I have a question. I know how to graph hyperbolas, ellipses, and parabolas on my calculator, but I don't know how to find the foci, directrix, or equation or the asymptotes on my calculator. Is there any way to find this without installing a program? I'm familiar with the conic section, I just don't know how to find the focus or directrix.

Answer : Many of the TI-84s come with an application. I believe that it's called conic sections or something to that effect. Look in the APPS section.