1.5 Conic sections  (Page 3/23)

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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.

A graph of a typical ellipse is shown in [link] . In this figure the foci are labeled as $F$ and ${F}^{\prime }.$ Both are the same fixed distance from the origin, and this distance is represented by the variable c . Therefore the coordinates of $F$ are $\left(c,0\right)$ and the coordinates of ${F}^{\prime }$ are $\left(\text{−}c,0\right).$ The points $P$ and ${P}^{\prime }$ are located at the ends of the major axis    of the ellipse, and have coordinates $\left(a,0\right)$ and $\left(\text{−}a,0\right),$ 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 ${P}^{\prime }$ are called the vertices of the ellipse. The points $Q$ and ${Q}^{\prime }$ are located at the ends of the minor axis    of the ellipse, and have coordinates $\left(0,b\right)$ and $\left(0,\text{−}b\right),$ 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 $\left(a,0\right),$ the sum of the distances is

$d\left(P,F\right)+d\left(P,{F}^{\prime }\right)=\left(a-c\right)+\left(a+c\right)=2a.$

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

$\begin{array}{ccc}\hfill d\left(A,F\right)+d\left(A,{F}^{\prime }\right)& =\hfill & 2a\hfill \\ \hfill \sqrt{{\left(x-c\right)}^{2}+{y}^{2}}+\sqrt{{\left(x+c\right)}^{2}+{y}^{2}}& =\hfill & 2a.\hfill \end{array}$

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

$\begin{array}{ccc}\hfill \sqrt{{\left(x-c\right)}^{2}+{y}^{2}}& =\hfill & 2a-\sqrt{{\left(x+c\right)}^{2}+{y}^{2}}\hfill \\ \hfill {\left(x-c\right)}^{2}+{y}^{2}& =\hfill & 4{a}^{2}-4a\sqrt{{\left(x+c\right)}^{2}+{y}^{2}}+{\left(x+c\right)}^{2}+{y}^{2}\hfill \\ \hfill {x}^{2}-2cx+{c}^{2}+{y}^{2}& =\hfill & 4{a}^{2}-4a\sqrt{{\left(x+c\right)}^{2}+{y}^{2}}+{x}^{2}+2cx+{c}^{2}+{y}^{2}\hfill \\ \hfill \text{−}2cx& =\hfill & 4{a}^{2}-4a\sqrt{{\left(x+c\right)}^{2}+{y}^{2}}+2cx.\hfill \end{array}$

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

$\begin{array}{ccc}\hfill -2cx& =\hfill & 4{a}^{2}-4a\sqrt{{\left(x+c\right)}^{2}+{y}^{2}}+2cx\hfill \\ \hfill 4a\sqrt{{\left(x+c\right)}^{2}+{y}^{2}}& =\hfill & 4{a}^{2}+4cx\hfill \\ \hfill \sqrt{{\left(x+c\right)}^{2}+{y}^{2}}& =\hfill & a+\frac{cx}{a}\hfill \\ \hfill {\left(x+c\right)}^{2}+{y}^{2}& =\hfill & {a}^{2}+2cx+\frac{{c}^{2}{x}^{2}}{{a}^{2}}\hfill \\ \hfill {x}^{2}+2cx+{c}^{2}+{y}^{2}& =\hfill & {a}^{2}+2cx+\frac{{c}^{2}{x}^{2}}{{a}^{2}}\hfill \\ \hfill {x}^{2}+{c}^{2}+{y}^{2}& =\hfill & {a}^{2}+\frac{{c}^{2}{x}^{2}}{{a}^{2}}.\hfill \end{array}$

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

$\begin{array}{}\\ \hfill {x}^{2}-\frac{{c}^{2}{x}^{2}}{{a}^{2}}+{y}^{2}& =\hfill & {a}^{2}-{c}^{2}\hfill \\ \hfill \frac{\left({a}^{2}-{c}^{2}\right){x}^{2}}{{a}^{2}}+{y}^{2}& =\hfill & {a}^{2}-{c}^{2}.\hfill \end{array}$

where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
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?
what is a peer
What is meant by 'nano scale'?
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
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
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?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
can you provide the details of the parametric equations for the lines that defince doubly-ruled surfeces (huperbolids of one sheet and hyperbolic paraboloid). Can you explain each of the variables in the equations?