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Introduction to Conic Sections

Conic sections (Latin: "Conus", "cone" and "sections") are curves generated by a plane cutting the sphere into two or more congruent pieces, the pieces being called "conies". The curve is named for its resemblance to a section of a cone. Conic sections are important in the study of geometry and can be found in the definitions of several different concepts, including conic sections, conics, conics in higher dimensions, and the conics of Apollonius. Conic sections can be classified into two kinds: "simple", consisting of a single curve and a single point; and "multiple", consisting of two or more curves and points. The simplest conic section is the circle, which is a plane curve of constant width (the radius). The next simplest is the ellipse, which is a plane curve of constant width (the minor axis) along which the points of the curve are collinear (parallel). The next simplest are the hyperbola and the parabola, which are plane curves of constant width (the major axis) in which the points are collinear (parallel). The last, and most complicated, is the hyperboloid of one sheet, which is a plane curve of constant width (the base) and constant height (the perpendicular height). All these curves are examples of conic sections in two dimensions. They can also be defined in higher dimensions, as curves that are generated by cutting a sphere along a plane. Conic sections have numerous applications in various fields, including optics, mechanical engineering and map projections.

Parabolas

489 Practice Problems
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04:07
Engineering Mechanics: Statics and Dynamics

Determine the moments of inertia $I_{x}$ and $I_{y}$ of the paraboloid of revolution. The mass of the paraboloid is 20 slug.

Three-Dimensional Kinetics of a Rigid Body
Shoukat Ali
03:06
Engineering Mechanics: Statics and Dynamics

The paraboloid is formed by revolving the shaded area around the $x$ axis. Determine the radius of gyration $k_{x}$ The density of the material is $\rho=5 \mathrm{Mg} / \mathrm{m}^{3}$.

Planar Kinetics of a Rigid Body: Force and Acceleration
Ahmed Kamel
07:03
Principles of Physics a Calculus Based Text

Review. One side of the roof of a house slopes up at $37.0^{\circ} .$ A roofer kicks a round, flat rock that has been thrown onto the roof by a neighborhood child. The rock slides straight up the incline with an initial speed of $15.0 \mathrm{m} / \mathrm{s}$. The coefficient of kinetic friction between the rock and the roof is 0.400 . The rock slides $10.0 \mathrm{m}$ up the roof to its peak. It crosses the ridge and goes into free fall, following a parabolic trajectory above the far side of the roof, with negligible air resistance. Determine the maximum height the rock reaches above the point where it was kicked.

More Applications of Newton's Laws
Vishal Gupta

Ellipses

498 Practice Problems
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00:45
21st Century Astronomy

Kepler's second law says that
a. planetary orbits are ellipses with the Sun at one focus.
b. the square of a planet's orbital period equals the cube of its semimajor axis.
c. net forces cause changes in motion.
d. planets move fastest when they are closest to the Sun.

Motion of Astronomical Bodies
Eve Rafferty
01:03
21st Century Astronomy

A lunar eclipse occurs when _______ shadow falls on _______.
a. Earth's; the Moon
b. the Moon's; Earth
c. the Sun's; the Moon
d. the Sun's; Earth

Patterns in the Sky-Motions of Earth and the Moon
Zachary Warner
01:04
21st Century Astronomy

Suppose a planet is discovered orbiting a star in a highly elliptical orbit. While the planet is close to the star it moves________, but while it is far away it moves________.
a. faster; slower
b. slower; faster
c. retrograde; prograde
d. prograde; retrograde

Motion of Astronomical Bodies
Eve Rafferty

Hyperbolas

398 Practice Problems
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00:17
Beginning and Intermediate Algebra

Suppose that a nonlinear system is composed of equations whose graphs are those described, and the number of points of intersection of the two graphs is as given. Make a sketch satisfying these conditions. (There may be more than one way to do this.)
A parabola and a hyperbola; two points

Nonlinear Functions, Conic Sections, and Nonlinear Systems
Nonlinear Systems of Equations
Vishal Parmar
07:23
Calculus: Early Transcendental Functions

Find an equation for the indicated conic section.
Hyperbola with foci (2,2) and (2,6) and vertices (2,3) and (2,5)

Parametric Equations and Polar Coordinates
Conic Sections
Kelly Brooks
00:23
Calculus: Early Transcendentals

What are the equations of the asymptotes of a standard hyperbola with vertices on the $x$ -axis?

Parametric and Polar Curves
Conic Sections
Grace Bajar

Conic Sections written in Polar Coordinates

244 Practice Problems
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19:01
Precalculus : Building Concepts and Connections

For each of the equations.
(a) Identify the conic section.
(b) Write the equation in the uv-coordinate system so that there is no uv term in the equation.
(c) Find the vertex, or vertices, in the uv-coordinate system, and graph.
$$x^{2}-10 \sqrt{3} x y+11 y^{2}-16=0$$

Conic Sections
Rotation of Axes; Ceneral Form of Conic Sections
Kevin Harmer
01:24
Precalculus : Building Concepts and Connections

Find the vertex, or vertices, of the given conic sections in the uv-coordinate system, obtained by rotating the $x-$and $y$ -axes by $\theta=\frac{\pi}{6}$.
$$4 u^{2}+v^{2}=36$$

Conic Sections
Rotation of Axes; Ceneral Form of Conic Sections
Kevin Harmer
02:46
Precalculus : Building Concepts and Connections

Identify the conic section given by each of the equations by using the general form of the conic equations.
$$x^{2}-2 x+y^{2}-1=0$$

Conic Sections
Rotation of Axes; Ceneral Form of Conic Sections
Deekshitha Balaji

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