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A parabola is drawn with vertex at the origin and opening up. Two parallel lines are drawn that strike the parabola and reflect to the focus.

Consider a parabolic dish designed to collect signals from a satellite in space. The dish is aimed directly at the satellite, and a receiver is located at the focus of the parabola. Radio waves coming in from the satellite are reflected off the surface of the parabola to the receiver, which collects and decodes the digital signals. This allows a small receiver to gather signals from a wide angle of sky. Flashlights and headlights in a car work on the same principle, but in reverse: the source of the light (that is, the light bulb) is located at the focus and the reflecting surface on the parabolic mirror focuses the beam straight ahead. This allows a small light bulb to illuminate a wide angle of space in front of the flashlight or car.

Ellipses

An ellipse can also be defined in terms of distances. In the case of an ellipse, there are two foci (plural of focus), and two directrices (plural of directrix). We look at the directrices in more detail later in this section.

Definition

An ellipse is the set of all points for which the sum of their distances from two fixed points (the foci) is constant.

An ellipse is drawn with center at the origin O, focal point F’ being (−c, 0) and focal point F being (c, 0). The ellipse has points P and P’ on the x axis and points Q and Q’ on the y axis. There are lines drawn from F’ to Q and F to Q. There are also lines drawn from F’ and F to a point A on the ellipse marked (x, y). The distance from O to Q and O to Q’ is marked b, and the distance from P to O and O to P’ is marked a.
A typical ellipse in which the sum of the distances from any point on the ellipse to the foci is constant.

A graph of a typical ellipse is shown in [link] . In this figure the foci are labeled as F and F . Both are the same fixed distance from the origin, and this distance is represented by the variable c . Therefore the coordinates of F are ( c , 0 ) and the coordinates of F are ( c , 0 ) . The points P and P are located at the ends of the major axis    of the ellipse, and have coordinates ( a , 0 ) and ( a , 0 ) , respectively. The major axis is always the longest distance across the ellipse, and can be horizontal or vertical. Thus, the length of the major axis in this ellipse is 2 a. Furthermore, P and P are called the vertices of the ellipse. The points Q and Q are located at the ends of the minor axis    of the ellipse, and have coordinates ( 0 , b ) and ( 0 , b ) , respectively. The minor axis is the shortest distance across the ellipse. The minor axis is perpendicular to the major axis.

According to the definition of the ellipse, we can choose any point on the ellipse and the sum of the distances from this point to the two foci is constant. Suppose we choose the point P. Since the coordinates of point P are ( a , 0 ) , the sum of the distances is

d ( P , F ) + d ( P , F ) = ( a c ) + ( a + c ) = 2 a .

Therefore the sum of the distances from an arbitrary point A with coordinates ( x , y ) is also equal to 2 a. Using the distance formula, we get

d ( A , F ) + d ( A , F ) = 2 a ( x c ) 2 + y 2 + ( x + c ) 2 + y 2 = 2 a .

Subtract the second radical from both sides and square both sides:

( x c ) 2 + y 2 = 2 a ( x + c ) 2 + y 2 ( x c ) 2 + y 2 = 4 a 2 4 a ( x + c ) 2 + y 2 + ( x + c ) 2 + y 2 x 2 2 c x + c 2 + y 2 = 4 a 2 4 a ( x + c ) 2 + y 2 + x 2 + 2 c x + c 2 + y 2 2 c x = 4 a 2 4 a ( x + c ) 2 + y 2 + 2 c x .

Now isolate the radical on the right-hand side and square again:

2 c x = 4 a 2 4 a ( x + c ) 2 + y 2 + 2 c x 4 a ( x + c ) 2 + y 2 = 4 a 2 + 4 c x ( x + c ) 2 + y 2 = a + c x a ( x + c ) 2 + y 2 = a 2 + 2 c x + c 2 x 2 a 2 x 2 + 2 c x + c 2 + y 2 = a 2 + 2 c x + c 2 x 2 a 2 x 2 + c 2 + y 2 = a 2 + c 2 x 2 a 2 .

Isolate the variables on the left-hand side of the equation and the constants on the right-hand side:

x 2 c 2 x 2 a 2 + y 2 = a 2 c 2 ( a 2 c 2 ) x 2 a 2 + y 2 = a 2 c 2 .

Questions & Answers

I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
How can I make nanorobot?
Lily
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Calculus volume 3. OpenStax CNX. Feb 05, 2016 Download for free at http://legacy.cnx.org/content/col11966/1.2
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