Question

Hyperbolas, Circles, Parabolas and Ellipses Tutorial!!!!

Answer 1

Definition of circle: Circle is the set of points that lies on given distance from the given point. The equation of circle with center at the origin in coordinate plane is x² + y² = R².
Definition of ellipse: Ellipse is the set of points, for which the sum of distances from them to the two given points, called foci, is constant and is bigger than distance between foci. The equation for ellipse, with the center at the origin in coordinate plane is x²/a² + y²/b²=1, where a is half of major axis and b is half of minor axis. If the ellipse has vertical orientation then its standard form equation is y²/a²+x²/b²=1.
Definition of parabola: Parabola is the set of points that lies on the same distance from the given line, called directrix, to the given point, called focus. The equation for the parabola with the vertex at the origin is y=2px² or x=2py², where p is the distance between focus and directrix.
Definition of hyperbola: Hyperbola is the set of points, for which the difference between them and two given point, called foci, is constant and is smaller than distance between foci. The equation for hyperbola in standard form is x²/a² – y²/b²=1, where a is the major axis of the hyperbola and b is minor axis. In vertical orientation the hyperbola has a standard form equation y²/a² – x²/b²=1.

Projective properties of hyperbola and ellipse:
1) If in ellipse the source of light is placed in one of the foci, the rays from this source will always intersect in second focus.
2) Light source placed in one of the foci of the hyperbola, will reflect from the branch of hyperbola, so that the reflected rays continuations will intersect in the second focus.

Asymptotes of hyperbola:
Hyperbola has two asymptotes. They are given the equations x/a+y/b=0 and x/a–y/b=0. In vertical orientation the asymptotes will be y/a+x/b=0 and y/a-x/b=0.

Eccentricities, ε:
Hyperbola: ε>1
Parabola: ε=1
Ellipse: 0<ε<1
Circle: ε=0

Foci:
Hyperbola: c²=a²+b², where c is the distance from center to focus.
Parabola: c=1/(4|a|), where c is the distance from vertex to focus.
Ellipse: c²=a² – b², where c is the distance form center to focus.
Circle: circle has no foci.

Uses and examples of hyperbolas, ellipses, circles and parabolas:
Hyperbolas are used in some constructions. Some examples of them in architecture are Shukhov Tower in Russia, Cathedral of Brasília in Brasilia, Gettysburg National Tower in USA.
During the World War II there were used hyperbolic navigation systems. When two pairs signals, given at the same time from the land by two towers and were received by the navigator, he measured the difference of time reached by the signals to him and constucted two hyberbolas based on this difference. The intersection of hyperbolas gave the coordinates of position of airplane or ship on map. The hyperbolic navigation systems are still used in navigations.
Circles are used nearly everywhere in everyday life. Some examples are car's wheel, round clock, coins.
Parabolas are used in some constructions, but rarely. The examples of it are the trajectory of water coming from fountain, trajectory of falling ball, paraboloic mirrors in some telescops.
Ellipses are used in mechanisms. The example of one well-known ellipse is the Earth's orbit around the Sun.

Some facts about conics and satelites:
The satelite moves around the Earth, with its first space speed.
The satelite has a paraboloic path, if it has second space speed. It will never return into the point where it was launched.
The satelite will have hyperbolic path, if is has third or higher space speeds.

Answer 2

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Answer 3

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Answer 4

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submit here

Answer 5

with your link

Answer 6

Well done

Answer 7

Good job :)

Answer 8

thanks

Answer 9

Nice Tutorial!

Answer 10

Cool :)