# 1.5 Conic sections  (Page 4/23)

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

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

If we refer back to [link] , then the length of each of the two green line segments is equal to a . This is true because the sum of the distances from the point Q to the foci $F\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}{F}^{\prime }$ is equal to 2 a , and the lengths of these two line segments are equal. This line segment forms a right triangle with hypotenuse length a and leg lengths b and c . From the Pythagorean theorem, ${a}^{2}+{b}^{2}={c}^{2}$ and ${b}^{2}={a}^{2}-{c}^{2}.$ Therefore the equation of the ellipse becomes

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

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

## Equation of an ellipse in standard form

Consider the ellipse with center $\left(h,k\right),$ a horizontal major axis with length 2 a , and a vertical minor axis with length 2 b . Then the equation of this ellipse in standard form 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 directrices are $x=h±\frac{{a}^{2}}{c}.$

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

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

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

If the major axis is horizontal, then the ellipse is called horizontal, and if the major axis is vertical, then the ellipse is called vertical. The equation of an ellipse is in general form if it is in the form $A{x}^{2}+B{y}^{2}+Cx+Dy+E=0,$ where A and B are either both positive or both negative. To convert the equation from general to standard form, use the method of completing the square.

## Finding the standard form of an ellipse

Put the equation $9{x}^{2}+4{y}^{2}-36x+24y+36=0$ into standard form and graph the resulting ellipse.

First subtract 36 from both sides of the equation:

$9{x}^{2}+4{y}^{2}-36x+24y=-36.$

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

$\begin{array}{ccc}\hfill \left(9{x}^{2}-36x\right)+\left(4{y}^{2}+24y\right)& =\hfill & -36\hfill \\ \hfill 9\left({x}^{2}-4x\right)+4\left({y}^{2}+6y\right)& =\hfill & -36.\hfill \end{array}$

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 ${\left(\frac{6}{2}\right)}^{2}=9.$ 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 adding 36 to the second set as well. Therefore the equation becomes

$\begin{array}{}\\ 9\left({x}^{2}-4x+4\right)+4\left({y}^{2}+6y+9\right)=-36+36+36\hfill \\ 9\left({x}^{2}-4x+4\right)+4\left({y}^{2}+6y+9\right)=36.\hfill \end{array}$

Now factor both sets of parentheses and divide by 36:

$\begin{array}{}\\ \hfill 9{\left(x-2\right)}^{2}+4{\left(y+3\right)}^{2}& =\hfill & 36\hfill \\ \hfill \frac{9{\left(x-2\right)}^{2}}{36}+\frac{4{\left(y+3\right)}^{2}}{36}& =\hfill & 1\hfill \\ \hfill \frac{{\left(x-2\right)}^{2}}{4}+\frac{{\left(y+3\right)}^{2}}{9}& =\hfill & 1.\hfill \end{array}$

The equation is now in standard form. Comparing this to [link] gives $h=2,$ $k=-3,$ $a=3,$ and $b=2.$ This is a vertical ellipse with center at $\left(2,-3\right),$ major axis 6, and minor axis 4. The graph of this ellipse appears as follows.

Put the equation $9{x}^{2}+16{y}^{2}+18x-64y-71=0$ into standard form and graph the resulting ellipse.

$\frac{{\left(x+1\right)}^{2}}{16}+\frac{{\left(y-2\right)}^{2}}{9}=1$

According to Kepler’s first law of planetary motion, the orbit of a planet around the Sun is an ellipse with the Sun at one of the foci as shown in [link] (a). Because Earth’s orbit is an ellipse, the distance from the Sun varies throughout the year. A commonly held misconception is that Earth is closer to the Sun in the summer. In fact, in summer for the northern hemisphere, Earth is farther from the Sun than during winter. The difference in season is caused by the tilt of Earth’s axis in the orbital plane. Comets that orbit the Sun, such as Halley’s Comet, also have elliptical orbits, as do moons orbiting the planets and satellites orbiting Earth.

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
learn
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 ?
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?