# 1.5 Conic sections  (Page 6/23)

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

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

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

Finally, divide both sides by ${a}^{2}-{c}^{2}.$ This gives the equation

$\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{a}^{2}-{c}^{2}}=1.$

We now define b so that ${b}^{2}={c}^{2}-{a}^{2}.$ This is possible because $c>a.$ Therefore the equation of the ellipse becomes

$\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1.$

Finally, if the center of the hyperbola is moved from the origin to the point $\left(h,k\right),$ we have the following standard form of a hyperbola.

## Equation of a hyperbola in standard form

Consider the hyperbola with center $\left(h,k\right),$ a horizontal major axis, and a vertical minor axis. Then the equation of this ellipse is

$\frac{{\left(x-h\right)}^{2}}{{a}^{2}}-\frac{{\left(y-k\right)}^{2}}{{b}^{2}}=1$

and the foci are located at $\left(h±c,k\right),$ where ${c}^{2}={a}^{2}+{b}^{2}.$ The equations of the asymptotes are given by $y=k±\frac{b}{a}\left(x-h\right).$ The equations of the directrices are

$x=k±\frac{{a}^{2}}{\sqrt{{a}^{2}+{b}^{2}}}=h±\frac{{a}^{2}}{c}.$

If the major axis is vertical, then the equation of the hyperbola becomes

$\frac{{\left(y-k\right)}^{2}}{{a}^{2}}-\frac{{\left(x-h\right)}^{2}}{{b}^{2}}=1$

and the foci are located at $\left(h,k±c\right),$ where ${c}^{2}={a}^{2}+{b}^{2}.$ The equations of the asymptotes are given by $y=k±\frac{a}{b}\left(x-h\right).$ The equations of the directrices are

$y=k±\frac{{a}^{2}}{\sqrt{{a}^{2}+{b}^{2}}}=k±\frac{{a}^{2}}{c}.$

If the major axis (transverse axis) is horizontal, then the hyperbola is called horizontal, and if the major axis is vertical then the hyperbola is called vertical. The equation of a hyperbola is in general form if it is in the form $A{x}^{2}+B{y}^{2}+Cx+Dy+E=0,$ where A and B have opposite signs. In order to convert the equation from general to standard form, use the method of completing the square.

## Finding the standard form of a hyperbola

Put the equation $9{x}^{2}-16{y}^{2}+36x+32y-124=0$ into standard form and graph the resulting hyperbola. What are the equations of the asymptotes?

First add 124 to both sides of the equation:

$9{x}^{2}-16{y}^{2}+36x+32y=124.$

Next group the x terms together and the y terms together, then factor out the common factors:

$\begin{array}{ccc}\hfill \left(9{x}^{2}+36x\right)-\left(16{y}^{2}-32y\right)& =\hfill & 124\hfill \\ \hfill 9\left({x}^{2}+4x\right)-16\left({y}^{2}-2y\right)& =\hfill & 124.\hfill \end{array}$

We need to determine the constant that, when added inside each set of parentheses, results in a perfect square. In the first set of parentheses, take half the coefficient of x and square it. This gives ${\left(\frac{4}{2}\right)}^{2}=4.$ In the second set of parentheses, take half the coefficient of y and square it. This gives ${\left(\frac{-2}{2}\right)}^{2}=1.$ Add these inside each pair of parentheses. Since the first set of parentheses has a 9 in front, we are actually adding 36 to the left-hand side. Similarly, we are subtracting 16 from the second set of parentheses. Therefore the equation becomes

$\begin{array}{}\\ 9\left({x}^{2}+4x+4\right)-16\left({y}^{2}-2y+1\right)=124+36-16\hfill \\ 9\left({x}^{2}+4x+4\right)-16\left({y}^{2}-2y+1\right)=144.\hfill \end{array}$

Next factor both sets of parentheses and divide by 144:

$\begin{array}{}\\ \hfill 9{\left(x+2\right)}^{2}-16{\left(y-1\right)}^{2}& =\hfill & 144\hfill \\ \hfill \frac{9{\left(x+2\right)}^{2}}{144}-\frac{16{\left(y-1\right)}^{2}}{144}& =\hfill & 1\hfill \\ \hfill \frac{{\left(x+2\right)}^{2}}{16}-\frac{{\left(y-1\right)}^{2}}{9}& =\hfill & 1.\hfill \end{array}$

The equation is now in standard form. Comparing this to [link] gives $h=-2,$ $k=1,$ $a=4,$ and $b=3.$ This is a horizontal hyperbola with center at $\left(-2,1\right)$ and asymptotes given by the equations $y=1±\frac{3}{4}\left(x+2\right).$ The graph of this hyperbola appears in the following figure.

Put the equation $4{y}^{2}-9{x}^{2}+16y+18x-29=0$ into standard form and graph the resulting hyperbola. What are the equations of the asymptotes?

$\frac{{\left(y+2\right)}^{2}}{9}-\frac{{\left(x-1\right)}^{2}}{4}=1.$ This is a vertical hyperbola. Asymptotes $y=-2±\frac{3}{2}\left(x-1\right).$

where we get a research paper on Nano chemistry....?
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
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da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
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 ?
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Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
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what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
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Good
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?